| L(s) = 1 | + 5·5-s + 3·7-s − 25·11-s + 29·13-s + 28·17-s + 64·19-s + 89·23-s − 139·25-s + 129·29-s + 241·31-s + 15·35-s + 366·37-s + 171·41-s + 803·43-s − 477·47-s − 317·49-s + 374·53-s − 125·55-s − 607·59-s + 1.34e3·61-s + 145·65-s + 1.54e3·67-s − 812·71-s + 1.92e3·73-s − 75·77-s + 1.72e3·79-s − 1.02e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.161·7-s − 0.685·11-s + 0.618·13-s + 0.399·17-s + 0.772·19-s + 0.806·23-s − 1.11·25-s + 0.826·29-s + 1.39·31-s + 0.0724·35-s + 1.62·37-s + 0.651·41-s + 2.84·43-s − 1.48·47-s − 0.924·49-s + 0.969·53-s − 0.306·55-s − 1.33·59-s + 2.83·61-s + 0.276·65-s + 2.82·67-s − 1.35·71-s + 3.09·73-s − 0.111·77-s + 2.45·79-s − 1.35·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.36259495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.36259495\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2 \wr S_5$ | \( 1 - p T + 164 T^{2} - 1403 T^{3} + 19551 T^{4} - 107972 T^{5} + 19551 p^{3} T^{6} - 1403 p^{6} T^{7} + 164 p^{9} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 3 T + 326 T^{2} + 51 p^{2} T^{3} + 182053 T^{4} - 1512396 T^{5} + 182053 p^{3} T^{6} + 51 p^{8} T^{7} + 326 p^{9} T^{8} - 3 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 25 T + 2765 T^{2} + 55618 T^{3} + 4854537 T^{4} + 57026683 T^{5} + 4854537 p^{3} T^{6} + 55618 p^{6} T^{7} + 2765 p^{9} T^{8} + 25 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 29 T + 5580 T^{2} - 298287 T^{3} + 14663967 T^{4} - 1018994904 T^{5} + 14663967 p^{3} T^{6} - 298287 p^{6} T^{7} + 5580 p^{9} T^{8} - 29 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 28 T + 15638 T^{2} - 162490 T^{3} + 119390505 T^{4} - 709949140 T^{5} + 119390505 p^{3} T^{6} - 162490 p^{6} T^{7} + 15638 p^{9} T^{8} - 28 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 64 T + 24162 T^{2} - 1390692 T^{3} + 287822613 T^{4} - 13275437016 T^{5} + 287822613 p^{3} T^{6} - 1390692 p^{6} T^{7} + 24162 p^{9} T^{8} - 64 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 89 T + 12470 T^{2} + 101209 T^{3} + 295894293 T^{4} - 25844704004 T^{5} + 295894293 p^{3} T^{6} + 101209 p^{6} T^{7} + 12470 p^{9} T^{8} - 89 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 129 T + 85852 T^{2} - 10022163 T^{3} + 3512273383 T^{4} - 332991273672 T^{5} + 3512273383 p^{3} T^{6} - 10022163 p^{6} T^{7} + 85852 p^{9} T^{8} - 129 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 241 T + 115290 T^{2} - 24773139 T^{3} + 6024358077 T^{4} - 1050481543752 T^{5} + 6024358077 p^{3} T^{6} - 24773139 p^{6} T^{7} + 115290 p^{9} T^{8} - 241 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 366 T + 89741 T^{2} - 112704 p T^{3} - 2324928194 T^{4} + 930179549628 T^{5} - 2324928194 p^{3} T^{6} - 112704 p^{7} T^{7} + 89741 p^{9} T^{8} - 366 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 171 T + 344827 T^{2} - 46045782 T^{3} + 47490564565 T^{4} - 4721690884257 T^{5} + 47490564565 p^{3} T^{6} - 46045782 p^{6} T^{7} + 344827 p^{9} T^{8} - 171 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 803 T + 540621 T^{2} - 237070422 T^{3} + 91840403913 T^{4} - 27309240011097 T^{5} + 91840403913 p^{3} T^{6} - 237070422 p^{6} T^{7} + 540621 p^{9} T^{8} - 803 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 477 T + 334270 T^{2} + 103354515 T^{3} + 40480140781 T^{4} + 11043214743492 T^{5} + 40480140781 p^{3} T^{6} + 103354515 p^{6} T^{7} + 334270 p^{9} T^{8} + 477 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 374 T + 501437 T^{2} - 127646720 T^{3} + 109236555390 T^{4} - 22186569121460 T^{5} + 109236555390 p^{3} T^{6} - 127646720 p^{6} T^{7} + 501437 p^{9} T^{8} - 374 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 607 T + 932981 T^{2} + 386297110 T^{3} + 344459738793 T^{4} + 106662664668085 T^{5} + 344459738793 p^{3} T^{6} + 386297110 p^{6} T^{7} + 932981 p^{9} T^{8} + 607 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 1349 T + 1245084 T^{2} - 876240411 T^{3} + 526068222087 T^{4} - 263230053776916 T^{5} + 526068222087 p^{3} T^{6} - 876240411 p^{6} T^{7} + 1245084 p^{9} T^{8} - 1349 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1549 T + 1931973 T^{2} - 1715972538 T^{3} + 1251380077113 T^{4} - 758027757282951 T^{5} + 1251380077113 p^{3} T^{6} - 1715972538 p^{6} T^{7} + 1931973 p^{9} T^{8} - 1549 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 812 T + 1459811 T^{2} + 919255376 T^{3} + 933440523354 T^{4} + 450012781468808 T^{5} + 933440523354 p^{3} T^{6} + 919255376 p^{6} T^{7} + 1459811 p^{9} T^{8} + 812 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 1928 T + 3064566 T^{2} - 3056842890 T^{3} + 2689492203753 T^{4} - 1766080796602620 T^{5} + 2689492203753 p^{3} T^{6} - 3056842890 p^{6} T^{7} + 3064566 p^{9} T^{8} - 1928 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 1727 T + 2239098 T^{2} - 1679744733 T^{3} + 1234959905901 T^{4} - 731388354331416 T^{5} + 1234959905901 p^{3} T^{6} - 1679744733 p^{6} T^{7} + 2239098 p^{9} T^{8} - 1727 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 1025 T + 2123078 T^{2} + 1688754731 T^{3} + 2163827783409 T^{4} + 1331143078302704 T^{5} + 2163827783409 p^{3} T^{6} + 1688754731 p^{6} T^{7} + 2123078 p^{9} T^{8} + 1025 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 2310 T + 3246853 T^{2} - 2201872296 T^{3} + 785324969794 T^{4} + 79350439969692 T^{5} + 785324969794 p^{3} T^{6} - 2201872296 p^{6} T^{7} + 3246853 p^{9} T^{8} - 2310 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 2875 T + 6451059 T^{2} - 10122640134 T^{3} + 13211511352101 T^{4} - 13660056218345217 T^{5} + 13211511352101 p^{3} T^{6} - 10122640134 p^{6} T^{7} + 6451059 p^{9} T^{8} - 2875 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.85652775255036518551549572979, −5.75789488173605486065169435552, −5.67166344247519187256401360833, −5.63929263771576616145648132057, −5.23894681416639679854797842884, −4.82628251683087474493083912980, −4.76253361196303770827556639790, −4.69990692229824436503471586288, −4.46295890265209441011667371247, −4.43375117240600795009696841357, −3.70841308824892816291534123479, −3.56268006568381485611844177374, −3.52474836390175716225769580572, −3.42576309115324979793636165728, −3.19901530076047260603952004370, −2.53624199966187479874098782990, −2.39727911852881652477850791702, −2.37444954348123779720200403292, −2.16051740811157992250796854787, −1.92417622211677267329442039962, −1.27551607915097804608283099619, −0.961708932992569776379382974742, −0.917025276919847616217778444617, −0.57338513987383176429241157455, −0.52096456403876507026435177298,
0.52096456403876507026435177298, 0.57338513987383176429241157455, 0.917025276919847616217778444617, 0.961708932992569776379382974742, 1.27551607915097804608283099619, 1.92417622211677267329442039962, 2.16051740811157992250796854787, 2.37444954348123779720200403292, 2.39727911852881652477850791702, 2.53624199966187479874098782990, 3.19901530076047260603952004370, 3.42576309115324979793636165728, 3.52474836390175716225769580572, 3.56268006568381485611844177374, 3.70841308824892816291534123479, 4.43375117240600795009696841357, 4.46295890265209441011667371247, 4.69990692229824436503471586288, 4.76253361196303770827556639790, 4.82628251683087474493083912980, 5.23894681416639679854797842884, 5.63929263771576616145648132057, 5.67166344247519187256401360833, 5.75789488173605486065169435552, 5.85652775255036518551549572979
