| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s − 5·11-s − 14-s + 16-s + 6·19-s − 3·20-s + 5·22-s + 2·23-s + 4·25-s + 28-s + 9·29-s − 3·31-s − 32-s − 3·35-s + 6·37-s − 6·38-s + 3·40-s − 6·41-s − 4·43-s − 5·44-s − 2·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 1.50·11-s − 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.670·20-s + 1.06·22-s + 0.417·23-s + 4/5·25-s + 0.188·28-s + 1.67·29-s − 0.538·31-s − 0.176·32-s − 0.507·35-s + 0.986·37-s − 0.973·38-s + 0.474·40-s − 0.937·41-s − 0.609·43-s − 0.753·44-s − 0.294·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8931231347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8931231347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−15.10748936546958, −14.51710397237304, −13.93063334861874, −13.16049179119683, −12.82034248791805, −11.99998688454848, −11.69310138761218, −11.24469116925814, −10.69696909700523, −10.07781884113208, −9.726799915484518, −8.807090124645495, −8.311982821990122, −7.953101755880341, −7.497667534746457, −6.991955598413048, −6.311448826572272, −5.256668888119065, −5.092460161819741, −4.263452149357432, −3.415092493193339, −2.965850291890681, −2.227274616692874, −1.153246679216762, −0.4456243241917586,
0.4456243241917586, 1.153246679216762, 2.227274616692874, 2.965850291890681, 3.415092493193339, 4.263452149357432, 5.092460161819741, 5.256668888119065, 6.311448826572272, 6.991955598413048, 7.497667534746457, 7.953101755880341, 8.311982821990122, 8.807090124645495, 9.726799915484518, 10.07781884113208, 10.69696909700523, 11.24469116925814, 11.69310138761218, 11.99998688454848, 12.82034248791805, 13.16049179119683, 13.93063334861874, 14.51710397237304, 15.10748936546958
