| L(s) = 1 | − 4.09e3·2-s + 5.31e5·3-s + 1.67e7·4-s + 5.90e8·5-s − 2.17e9·6-s + 5.78e10·7-s − 6.87e10·8-s + 2.82e11·9-s − 2.41e12·10-s + 9.49e12·11-s + 8.91e12·12-s − 1.34e14·13-s − 2.36e14·14-s + 3.13e14·15-s + 2.81e14·16-s − 2.52e15·17-s − 1.15e15·18-s + 1.14e16·19-s + 9.90e15·20-s + 3.07e16·21-s − 3.88e16·22-s + 1.13e17·23-s − 3.65e16·24-s + 5.05e16·25-s + 5.52e17·26-s + 1.50e17·27-s + 9.70e17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.08·5-s − 0.408·6-s + 1.57·7-s − 0.353·8-s + 1/3·9-s − 0.764·10-s + 0.912·11-s + 0.288·12-s − 1.60·13-s − 1.11·14-s + 0.624·15-s + 1/4·16-s − 1.05·17-s − 0.235·18-s + 1.18·19-s + 0.540·20-s + 0.912·21-s − 0.644·22-s + 1.07·23-s − 0.204·24-s + 0.169·25-s + 1.13·26-s + 0.192·27-s + 0.789·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(2.511538135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511538135\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{12} T \) |
| 3 | \( 1 - p^{12} T \) |
| good | 5 | \( 1 - 590425734 T + p^{25} T^{2} \) |
| 7 | \( 1 - 1180763624 p^{2} T + p^{25} T^{2} \) |
| 11 | \( 1 - 863115112740 p T + p^{25} T^{2} \) |
| 13 | \( 1 + 10382155466266 p T + p^{25} T^{2} \) |
| 17 | \( 1 + 148594942164942 p T + p^{25} T^{2} \) |
| 19 | \( 1 - 603618888013724 p T + p^{25} T^{2} \) |
| 23 | \( 1 - 4927940469652200 p T + p^{25} T^{2} \) |
| 29 | \( 1 - 1081348899350530974 T + p^{25} T^{2} \) |
| 31 | \( 1 - 4649090467326833408 T + p^{25} T^{2} \) |
| 37 | \( 1 + 46093370056702003258 T + p^{25} T^{2} \) |
| 41 | \( 1 - 51449233931826001194 T + p^{25} T^{2} \) |
| 43 | \( 1 + \)\(36\!\cdots\!84\)\( T + p^{25} T^{2} \) |
| 47 | \( 1 + 49106637730499080080 T + p^{25} T^{2} \) |
| 53 | \( 1 - \)\(44\!\cdots\!34\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 - \)\(22\!\cdots\!40\)\( T + p^{25} T^{2} \) |
| 61 | \( 1 + \)\(12\!\cdots\!90\)\( T + p^{25} T^{2} \) |
| 67 | \( 1 - \)\(63\!\cdots\!96\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 - \)\(60\!\cdots\!20\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 + \)\(26\!\cdots\!58\)\( T + p^{25} T^{2} \) |
| 79 | \( 1 + \)\(26\!\cdots\!92\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 - \)\(68\!\cdots\!32\)\( T + p^{25} T^{2} \) |
| 89 | \( 1 + \)\(25\!\cdots\!42\)\( T + p^{25} T^{2} \) |
| 97 | \( 1 - \)\(25\!\cdots\!82\)\( T + p^{25} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.19646444743910735216830124745, −14.96415955236450470177651326971, −13.89839534676381886827868620781, −11.70248202416348403377744573134, −9.950796970381991117094593395033, −8.704013350856284948475673897030, −7.10946452830944605286771631441, −4.95711289645716974510611124583, −2.38181198715947332480137372555, −1.31318055944919469539949734514,
1.31318055944919469539949734514, 2.38181198715947332480137372555, 4.95711289645716974510611124583, 7.10946452830944605286771631441, 8.704013350856284948475673897030, 9.950796970381991117094593395033, 11.70248202416348403377744573134, 13.89839534676381886827868620781, 14.96415955236450470177651326971, 17.19646444743910735216830124745
