| L(s) = 1 | − 0.0242·2-s − 1.99·4-s + 2.24·5-s − 4.21·7-s + 0.0969·8-s − 0.0544·10-s + 2.12·11-s − 3.04·13-s + 0.102·14-s + 3.99·16-s + 4.57·17-s − 0.932·19-s − 4.48·20-s − 0.0515·22-s + 4.32·23-s + 0.0373·25-s + 0.0739·26-s + 8.42·28-s + 5.55·29-s − 1.27·31-s − 0.290·32-s − 0.110·34-s − 9.46·35-s − 8.37·37-s + 0.0226·38-s + 0.217·40-s − 10.8·41-s + ⋯ |
| L(s) = 1 | − 0.0171·2-s − 0.999·4-s + 1.00·5-s − 1.59·7-s + 0.0342·8-s − 0.0172·10-s + 0.640·11-s − 0.845·13-s + 0.0273·14-s + 0.999·16-s + 1.10·17-s − 0.213·19-s − 1.00·20-s − 0.0109·22-s + 0.902·23-s + 0.00747·25-s + 0.0144·26-s + 1.59·28-s + 1.03·29-s − 0.228·31-s − 0.0514·32-s − 0.0190·34-s − 1.59·35-s − 1.37·37-s + 0.00366·38-s + 0.0344·40-s − 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0242T + 2T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 + 0.932T + 19T^{2} \) |
| 23 | \( 1 - 4.32T + 23T^{2} \) |
| 29 | \( 1 - 5.55T + 29T^{2} \) |
| 31 | \( 1 + 1.27T + 31T^{2} \) |
| 37 | \( 1 + 8.37T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.30T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 3.59T + 97T^{2} \) |
| 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
−9.114929991666507758409761411460, −8.033652772777948517371381843972, −6.93038574457889648864161901258, −6.34091235094143457305772225221, −5.48361506391999639877438303807, −4.79214004704421302960095446283, −3.56099130262708696090516007884, −2.97965691777691923099123598416, −1.47315085461701331852602366454, 0,
1.47315085461701331852602366454, 2.97965691777691923099123598416, 3.56099130262708696090516007884, 4.79214004704421302960095446283, 5.48361506391999639877438303807, 6.34091235094143457305772225221, 6.93038574457889648864161901258, 8.033652772777948517371381843972, 9.114929991666507758409761411460
