| L(s) = 1 | − 2·5-s + 26·7-s + 52·11-s − 13·13-s − 48·17-s − 18·19-s − 52·23-s − 121·25-s − 224·29-s − 310·31-s − 52·35-s − 18·37-s − 330·41-s − 328·43-s + 616·47-s + 333·49-s + 324·53-s − 104·55-s + 188·59-s − 110·61-s + 26·65-s − 118·67-s − 656·71-s − 178·73-s + 1.35e3·77-s − 836·79-s + 60·83-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 1.40·7-s + 1.42·11-s − 0.277·13-s − 0.684·17-s − 0.217·19-s − 0.471·23-s − 0.967·25-s − 1.43·29-s − 1.79·31-s − 0.251·35-s − 0.0799·37-s − 1.25·41-s − 1.16·43-s + 1.91·47-s + 0.970·49-s + 0.839·53-s − 0.254·55-s + 0.414·59-s − 0.230·61-s + 0.0496·65-s − 0.215·67-s − 1.09·71-s − 0.285·73-s + 2.00·77-s − 1.19·79-s + 0.0793·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 17 | \( 1 + 48 T + p^{3} T^{2} \) |
| 19 | \( 1 + 18 T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 18 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 328 T + p^{3} T^{2} \) |
| 47 | \( 1 - 616 T + p^{3} T^{2} \) |
| 53 | \( 1 - 324 T + p^{3} T^{2} \) |
| 59 | \( 1 - 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 118 T + p^{3} T^{2} \) |
| 71 | \( 1 + 656 T + p^{3} T^{2} \) |
| 73 | \( 1 + 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 836 T + p^{3} T^{2} \) |
| 83 | \( 1 - 60 T + p^{3} T^{2} \) |
| 89 | \( 1 + 870 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1238 T + p^{3} T^{2} \) |
| 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
−8.643992876523902442611113240197, −7.59618940225602408069766563785, −7.10073320390877952595489273174, −6.00739686308708283489188058621, −5.21983098196027647162544431584, −4.23119392324671971842701578744, −3.70683617465845295600784785410, −2.07214033757791891576802956954, −1.51235117964480035869291468802, 0,
1.51235117964480035869291468802, 2.07214033757791891576802956954, 3.70683617465845295600784785410, 4.23119392324671971842701578744, 5.21983098196027647162544431584, 6.00739686308708283489188058621, 7.10073320390877952595489273174, 7.59618940225602408069766563785, 8.643992876523902442611113240197
