| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2.37·7-s − 8-s + 9-s + 6.37·11-s − 12-s + 2·13-s − 2.37·14-s + 16-s − 6.74·17-s − 18-s − 6.37·19-s − 2.37·21-s − 6.37·22-s + 2.37·23-s + 24-s − 2·26-s − 27-s + 2.37·28-s + 2.74·29-s − 31-s − 32-s − 6.37·33-s + 6.74·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.896·7-s − 0.353·8-s + 0.333·9-s + 1.92·11-s − 0.288·12-s + 0.554·13-s − 0.634·14-s + 0.250·16-s − 1.63·17-s − 0.235·18-s − 1.46·19-s − 0.517·21-s − 1.35·22-s + 0.494·23-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.448·28-s + 0.509·29-s − 0.179·31-s − 0.176·32-s − 1.10·33-s + 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 2.37T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 0.744T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−8.187739174110367421553311030345, −6.94043384359763290796180965102, −6.68804144562419707163809570057, −6.05975329508170626646811782590, −4.86624348356107088704035931927, −4.32630470800383424636289952301, −3.39198590475853754033954296876, −1.85955530219704585871673093711, −1.48270525370762976810609932868, 0,
1.48270525370762976810609932868, 1.85955530219704585871673093711, 3.39198590475853754033954296876, 4.32630470800383424636289952301, 4.86624348356107088704035931927, 6.05975329508170626646811782590, 6.68804144562419707163809570057, 6.94043384359763290796180965102, 8.187739174110367421553311030345
