| L(s) = 1 | + 6·3-s − 25·5-s + 15·7-s − 10·9-s + 153·11-s + 28·13-s − 150·15-s − 341·17-s − 3·19-s + 90·21-s − 115·23-s + 375·25-s − 105·27-s − 583·29-s − 662·31-s + 918·33-s − 375·35-s − 172·37-s + 168·39-s + 344·41-s + 230·43-s + 250·45-s + 337·47-s − 747·49-s − 2.04e3·51-s − 942·53-s − 3.82e3·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 2.23·5-s + 0.809·7-s − 0.370·9-s + 4.19·11-s + 0.597·13-s − 2.58·15-s − 4.86·17-s − 0.0362·19-s + 0.935·21-s − 1.04·23-s + 3·25-s − 0.748·27-s − 3.73·29-s − 3.83·31-s + 4.84·33-s − 1.81·35-s − 0.764·37-s + 0.689·39-s + 1.31·41-s + 0.815·43-s + 0.828·45-s + 1.04·47-s − 2.17·49-s − 5.61·51-s − 2.44·53-s − 9.37·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{5} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 - 2 p T + 46 T^{2} - 77 p T^{3} + 1421 T^{4} - 4621 T^{5} + 1421 p^{3} T^{6} - 77 p^{7} T^{7} + 46 p^{9} T^{8} - 2 p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - 15 T + 972 T^{2} - 13862 T^{3} + 446891 T^{4} - 6154870 T^{5} + 446891 p^{3} T^{6} - 13862 p^{6} T^{7} + 972 p^{9} T^{8} - 15 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 153 T + 13568 T^{2} - 842244 T^{3} + 41118259 T^{4} - 149259130 p T^{5} + 41118259 p^{3} T^{6} - 842244 p^{6} T^{7} + 13568 p^{9} T^{8} - 153 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 28 T + 5130 T^{2} - 144233 T^{3} + 14684355 T^{4} - 405986785 T^{5} + 14684355 p^{3} T^{6} - 144233 p^{6} T^{7} + 5130 p^{9} T^{8} - 28 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 341 T + 63482 T^{2} + 8153248 T^{3} + 799258521 T^{4} + 62382425942 T^{5} + 799258521 p^{3} T^{6} + 8153248 p^{6} T^{7} + 63482 p^{9} T^{8} + 341 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 3 T + 23814 T^{2} - 116088 T^{3} + 254475577 T^{4} - 1869204142 T^{5} + 254475577 p^{3} T^{6} - 116088 p^{6} T^{7} + 23814 p^{9} T^{8} + 3 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 583 T + 218618 T^{2} + 56606037 T^{3} + 11861518867 T^{4} + 2004352275864 T^{5} + 11861518867 p^{3} T^{6} + 56606037 p^{6} T^{7} + 218618 p^{9} T^{8} + 583 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 662 T + 242520 T^{2} + 59788509 T^{3} + 11718321225 T^{4} + 2046163869659 T^{5} + 11718321225 p^{3} T^{6} + 59788509 p^{6} T^{7} + 242520 p^{9} T^{8} + 662 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 172 T + 117849 T^{2} + 17932680 T^{3} + 9705307890 T^{4} + 1271277488696 T^{5} + 9705307890 p^{3} T^{6} + 17932680 p^{6} T^{7} + 117849 p^{9} T^{8} + 172 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 344 T + 246686 T^{2} - 60239383 T^{3} + 28247114415 T^{4} - 5589907146851 T^{5} + 28247114415 p^{3} T^{6} - 60239383 p^{6} T^{7} + 246686 p^{9} T^{8} - 344 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 230 T + 183803 T^{2} - 22637032 T^{3} + 16427067478 T^{4} - 970309895108 T^{5} + 16427067478 p^{3} T^{6} - 22637032 p^{6} T^{7} + 183803 p^{9} T^{8} - 230 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 337 T + 188924 T^{2} - 40884323 T^{3} + 29011021441 T^{4} - 7306837730572 T^{5} + 29011021441 p^{3} T^{6} - 40884323 p^{6} T^{7} + 188924 p^{9} T^{8} - 337 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 942 T + 847913 T^{2} + 467814984 T^{3} + 247735334338 T^{4} + 96466603775732 T^{5} + 247735334338 p^{3} T^{6} + 467814984 p^{6} T^{7} + 847913 p^{9} T^{8} + 942 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 1166 T + 1321491 T^{2} - 938983912 T^{3} + 593816946686 T^{4} - 286883407447412 T^{5} + 593816946686 p^{3} T^{6} - 938983912 p^{6} T^{7} + 1321491 p^{9} T^{8} - 1166 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 499 T + 899502 T^{2} - 368354140 T^{3} + 360389393101 T^{4} - 115747812047290 T^{5} + 360389393101 p^{3} T^{6} - 368354140 p^{6} T^{7} + 899502 p^{9} T^{8} - 499 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 972 T + 951975 T^{2} - 874847080 T^{3} + 574815613850 T^{4} - 330625329906584 T^{5} + 574815613850 p^{3} T^{6} - 874847080 p^{6} T^{7} + 951975 p^{9} T^{8} - 972 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 14 T + 1455746 T^{2} + 5910247 T^{3} + 937827920553 T^{4} + 7561146837177 T^{5} + 937827920553 p^{3} T^{6} + 5910247 p^{6} T^{7} + 1455746 p^{9} T^{8} - 14 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 229 T + 1181626 T^{2} + 380814759 T^{3} + 675019783655 T^{4} + 230923153684560 T^{5} + 675019783655 p^{3} T^{6} + 380814759 p^{6} T^{7} + 1181626 p^{9} T^{8} + 229 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 88 T + 1420035 T^{2} - 441701544 T^{3} + 975079710690 T^{4} - 369855406494592 T^{5} + 975079710690 p^{3} T^{6} - 441701544 p^{6} T^{7} + 1420035 p^{9} T^{8} - 88 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 72 T + 1925299 T^{2} - 221002912 T^{3} + 1723983623478 T^{4} - 208154794731184 T^{5} + 1723983623478 p^{3} T^{6} - 221002912 p^{6} T^{7} + 1925299 p^{9} T^{8} - 72 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 90 T + 1035961 T^{2} - 265414184 T^{3} + 788112824622 T^{4} - 183772332654020 T^{5} + 788112824622 p^{3} T^{6} - 265414184 p^{6} T^{7} + 1035961 p^{9} T^{8} + 90 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 1765 T + 5492684 T^{2} + 6586255418 T^{3} + 10876963650931 T^{4} + 9081690303986842 T^{5} + 10876963650931 p^{3} T^{6} + 6586255418 p^{6} T^{7} + 5492684 p^{9} T^{8} + 1765 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.46039513184575811468507215643, −5.25132577962139322621930977378, −5.16900010558051891948739891938, −5.03262362698262697409242249304, −5.02598272777065538206448375881, −4.45338729665467759634833011726, −4.29462108575745259659382322766, −4.18009547961842036849545186850, −4.06200600530621846045628557039, −3.93797604475647248610914161579, −3.93223586483696102654833428881, −3.55671015467560616689016475315, −3.50661999354755039717743928870, −3.48817576966031600535302982527, −3.48610575703025568205920044183, −2.53824597120655175506710748741, −2.47852060462931263075991599772, −2.41446826898233344573958817594, −2.40765664386518242106712313020, −1.96781809055002949656055140403, −1.67028228349539981585991990580, −1.63208551721476281378350141769, −1.28081591714383996932100623585, −1.16347957399957449153574169144, −0.985991722290076937342190312917, 0, 0, 0, 0, 0,
0.985991722290076937342190312917, 1.16347957399957449153574169144, 1.28081591714383996932100623585, 1.63208551721476281378350141769, 1.67028228349539981585991990580, 1.96781809055002949656055140403, 2.40765664386518242106712313020, 2.41446826898233344573958817594, 2.47852060462931263075991599772, 2.53824597120655175506710748741, 3.48610575703025568205920044183, 3.48817576966031600535302982527, 3.50661999354755039717743928870, 3.55671015467560616689016475315, 3.93223586483696102654833428881, 3.93797604475647248610914161579, 4.06200600530621846045628557039, 4.18009547961842036849545186850, 4.29462108575745259659382322766, 4.45338729665467759634833011726, 5.02598272777065538206448375881, 5.03262362698262697409242249304, 5.16900010558051891948739891938, 5.25132577962139322621930977378, 5.46039513184575811468507215643
