| L(s) = 1 | + 4·2-s + 13·3-s − 3·4-s + 11·5-s + 52·6-s + 24·7-s − 34·8-s + 40·9-s + 44·10-s + 61·11-s − 39·12-s − 37·13-s + 96·14-s + 143·15-s − 101·16-s + 130·17-s + 160·18-s − 22·19-s − 33·20-s + 312·21-s + 244·22-s + 73·23-s − 442·24-s − 239·25-s − 148·26-s − 172·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.50·3-s − 3/8·4-s + 0.983·5-s + 3.53·6-s + 1.29·7-s − 1.50·8-s + 1.48·9-s + 1.39·10-s + 1.67·11-s − 0.938·12-s − 0.789·13-s + 1.83·14-s + 2.46·15-s − 1.57·16-s + 1.85·17-s + 2.09·18-s − 0.265·19-s − 0.368·20-s + 3.24·21-s + 2.36·22-s + 0.661·23-s − 3.75·24-s − 1.91·25-s − 1.11·26-s − 1.22·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69343957 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69343957 ^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.13753365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.13753365\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 37 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - p^{2} T + 19 T^{2} - 27 p T^{3} + 119 p T^{4} - 81 p^{3} T^{5} + 119 p^{4} T^{6} - 27 p^{7} T^{7} + 19 p^{9} T^{8} - p^{14} T^{9} + p^{15} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 - 13 T + 43 p T^{2} - 985 T^{3} + 6679 T^{4} - 12368 p T^{5} + 6679 p^{3} T^{6} - 985 p^{6} T^{7} + 43 p^{10} T^{8} - 13 p^{12} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 11 T + 72 p T^{2} - 1982 T^{3} + 50831 T^{4} - 149678 T^{5} + 50831 p^{3} T^{6} - 1982 p^{6} T^{7} + 72 p^{10} T^{8} - 11 p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 24 T + 1620 T^{2} - 27570 T^{3} + 1047043 T^{4} - 1891300 p T^{5} + 1047043 p^{3} T^{6} - 27570 p^{6} T^{7} + 1620 p^{9} T^{8} - 24 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 61 T + 5051 T^{2} - 255845 T^{3} + 12951439 T^{4} - 459792952 T^{5} + 12951439 p^{3} T^{6} - 255845 p^{6} T^{7} + 5051 p^{9} T^{8} - 61 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 37 T + 6280 T^{2} + 210810 T^{3} + 20585759 T^{4} + 562017090 T^{5} + 20585759 p^{3} T^{6} + 210810 p^{6} T^{7} + 6280 p^{9} T^{8} + 37 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 130 T + 16993 T^{2} - 854528 T^{3} + 51563214 T^{4} - 996948732 T^{5} + 51563214 p^{3} T^{6} - 854528 p^{6} T^{7} + 16993 p^{9} T^{8} - 130 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 22 T + 19155 T^{2} - 511376 T^{3} + 137508950 T^{4} - 9154678956 T^{5} + 137508950 p^{3} T^{6} - 511376 p^{6} T^{7} + 19155 p^{9} T^{8} + 22 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 73 T + 7350 T^{2} - 1172382 T^{3} + 73498989 T^{4} + 7921444038 T^{5} + 73498989 p^{3} T^{6} - 1172382 p^{6} T^{7} + 7350 p^{9} T^{8} - 73 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 271 T + 3136 p T^{2} - 18197622 T^{3} + 4034991767 T^{4} - 610709165590 T^{5} + 4034991767 p^{3} T^{6} - 18197622 p^{6} T^{7} + 3136 p^{10} T^{8} - 271 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 363 T + 177480 T^{2} - 40997526 T^{3} + 11064003139 T^{4} - 1787158274590 T^{5} + 11064003139 p^{3} T^{6} - 40997526 p^{6} T^{7} + 177480 p^{9} T^{8} - 363 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 381 T + 246949 T^{2} - 68583035 T^{3} + 30018497091 T^{4} - 6596205610814 T^{5} + 30018497091 p^{3} T^{6} - 68583035 p^{6} T^{7} + 246949 p^{9} T^{8} - 381 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 + 408 T + 216779 T^{2} + 100229256 T^{3} + 33176381926 T^{4} + 9813782430304 T^{5} + 33176381926 p^{3} T^{6} + 100229256 p^{6} T^{7} + 216779 p^{9} T^{8} + 408 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 276 T + 497396 T^{2} - 110931710 T^{3} + 101119454099 T^{4} - 17095798030508 T^{5} + 101119454099 p^{3} T^{6} - 110931710 p^{6} T^{7} + 497396 p^{9} T^{8} - 276 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 156 T + 187042 T^{2} + 53751814 T^{3} + 18661712221 T^{4} + 10746004250972 T^{5} + 18661712221 p^{3} T^{6} + 53751814 p^{6} T^{7} + 187042 p^{9} T^{8} - 156 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 100 T + 654927 T^{2} + 10664920 T^{3} + 185098747034 T^{4} + 13016317564184 T^{5} + 185098747034 p^{3} T^{6} + 10664920 p^{6} T^{7} + 654927 p^{9} T^{8} - 100 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 1711 T + 1805688 T^{2} + 1373086042 T^{3} + 871087526715 T^{4} + 452288372272534 T^{5} + 871087526715 p^{3} T^{6} + 1373086042 p^{6} T^{7} + 1805688 p^{9} T^{8} + 1711 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 787 T + 620336 T^{2} - 406814856 T^{3} + 332115969971 T^{4} - 171999519830138 T^{5} + 332115969971 p^{3} T^{6} - 406814856 p^{6} T^{7} + 620336 p^{9} T^{8} - 787 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 1578 T + 1787320 T^{2} - 1495145048 T^{3} + 1203604171527 T^{4} - 771467517967436 T^{5} + 1203604171527 p^{3} T^{6} - 1495145048 p^{6} T^{7} + 1787320 p^{9} T^{8} - 1578 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 313 T + 1220937 T^{2} + 328617115 T^{3} + 768888241687 T^{4} + 185594332932058 T^{5} + 768888241687 p^{3} T^{6} + 328617115 p^{6} T^{7} + 1220937 p^{9} T^{8} + 313 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 569 T + 2257366 T^{2} - 1087160174 T^{3} + 2132614542713 T^{4} - 794961824309426 T^{5} + 2132614542713 p^{3} T^{6} - 1087160174 p^{6} T^{7} + 2257366 p^{9} T^{8} - 569 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 2422 T + 4413584 T^{2} - 5510916716 T^{3} + 5733083694659 T^{4} - 4709778205772604 T^{5} + 5733083694659 p^{3} T^{6} - 5510916716 p^{6} T^{7} + 4413584 p^{9} T^{8} - 2422 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 2466 T + 4850165 T^{2} + 6850335912 T^{3} + 7679772580354 T^{4} + 7205242275163468 T^{5} + 7679772580354 p^{3} T^{6} + 6850335912 p^{6} T^{7} + 4850165 p^{9} T^{8} + 2466 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 2406 T + 5214689 T^{2} + 7623508800 T^{3} + 9725697363454 T^{4} + 10015485558497588 T^{5} + 9725697363454 p^{3} T^{6} + 7623508800 p^{6} T^{7} + 5214689 p^{9} T^{8} + 2406 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08976662984497465131658886225, −9.575687755208576860480279575043, −9.448568830873263343152692125698, −9.378793286570060407164660645748, −9.250110285241939407310010730642, −8.501289119883936690031069929273, −8.498498073222671690724363421669, −8.232983642415302491331342244425, −8.221993368651064041117971735082, −7.60721700948092894492904470373, −7.59817395508620864325194153323, −6.74176553035864309701048325543, −6.53556218705602231878724449322, −6.18335410772384154910714091429, −5.89894839889632647515446775532, −5.16691762748347530989101048229, −5.01821854609445596811497325013, −4.81312500994488368425117293648, −4.35705744866897044148136680398, −3.88352132058545875777878828197, −3.52737783387336051922432317502, −3.04883537401242207007917974426, −2.66543970646809800614893189240, −2.12383503890829448781329682417, −1.37111196387837379888456996989,
1.37111196387837379888456996989, 2.12383503890829448781329682417, 2.66543970646809800614893189240, 3.04883537401242207007917974426, 3.52737783387336051922432317502, 3.88352132058545875777878828197, 4.35705744866897044148136680398, 4.81312500994488368425117293648, 5.01821854609445596811497325013, 5.16691762748347530989101048229, 5.89894839889632647515446775532, 6.18335410772384154910714091429, 6.53556218705602231878724449322, 6.74176553035864309701048325543, 7.59817395508620864325194153323, 7.60721700948092894492904470373, 8.221993368651064041117971735082, 8.232983642415302491331342244425, 8.498498073222671690724363421669, 8.501289119883936690031069929273, 9.250110285241939407310010730642, 9.378793286570060407164660645748, 9.448568830873263343152692125698, 9.575687755208576860480279575043, 10.08976662984497465131658886225
