| L(s) = 1 | + 2·2-s − 2·3-s + 4·4-s + 5·5-s − 4·6-s + 2·7-s + 8·8-s − 23·9-s + 10·10-s − 72·11-s − 8·12-s + 2·13-s + 4·14-s − 10·15-s + 16·16-s − 66·17-s − 46·18-s + 38·19-s + 20·20-s − 4·21-s − 144·22-s − 36·23-s − 16·24-s + 25·25-s + 4·26-s + 100·27-s + 8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.447·5-s − 0.272·6-s + 0.107·7-s + 0.353·8-s − 0.851·9-s + 0.316·10-s − 1.97·11-s − 0.192·12-s + 0.0426·13-s + 0.0763·14-s − 0.172·15-s + 1/4·16-s − 0.941·17-s − 0.602·18-s + 0.458·19-s + 0.223·20-s − 0.0415·21-s − 1.39·22-s − 0.326·23-s − 0.136·24-s + 1/5·25-s + 0.0301·26-s + 0.712·27-s + 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 37 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 - 272 T + p^{3} T^{2} \) |
| 47 | \( 1 + 198 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 + 498 T + p^{3} T^{2} \) |
| 61 | \( 1 - 542 T + p^{3} T^{2} \) |
| 67 | \( 1 - 2 T + p^{3} T^{2} \) |
| 71 | \( 1 - 408 T + p^{3} T^{2} \) |
| 73 | \( 1 + 358 T + p^{3} T^{2} \) |
| 79 | \( 1 - 722 T + p^{3} T^{2} \) |
| 83 | \( 1 + 174 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 574 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80810442129439426518219086134, −9.823102597670642768971803319129, −8.518645895854152572270613085106, −7.59693975892503189190290377608, −6.37659250238105165878514762441, −5.46325635215664158764464876927, −4.84454015462948887129197984873, −3.17766655495608365410665524116, −2.14680185405100477334036135518, 0,
2.14680185405100477334036135518, 3.17766655495608365410665524116, 4.84454015462948887129197984873, 5.46325635215664158764464876927, 6.37659250238105165878514762441, 7.59693975892503189190290377608, 8.518645895854152572270613085106, 9.823102597670642768971803319129, 10.80810442129439426518219086134
