| L(s) = 1 | − 2-s − 3-s + 4-s + 2.22·5-s + 6-s − 0.652·7-s − 8-s + 9-s − 2.22·10-s − 1.53·11-s − 12-s − 0.467·13-s + 0.652·14-s − 2.22·15-s + 16-s − 2.10·17-s − 18-s + 2.22·20-s + 0.652·21-s + 1.53·22-s + 5.90·23-s + 24-s − 0.0418·25-s + 0.467·26-s − 27-s − 0.652·28-s − 8.33·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.995·5-s + 0.408·6-s − 0.246·7-s − 0.353·8-s + 0.333·9-s − 0.704·10-s − 0.461·11-s − 0.288·12-s − 0.129·13-s + 0.174·14-s − 0.574·15-s + 0.250·16-s − 0.510·17-s − 0.235·18-s + 0.497·20-s + 0.142·21-s + 0.326·22-s + 1.23·23-s + 0.204·24-s − 0.00837·25-s + 0.0917·26-s − 0.192·27-s − 0.123·28-s − 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 + 0.652T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 + 0.467T + 13T^{2} \) |
| 17 | \( 1 + 2.10T + 17T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 + 8.63T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 0.248T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 3.70T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−9.084375042641264038456207040056, −7.80649049598333949380109845076, −7.15924808448895041647147210291, −6.33716156265809256001812396913, −5.63901990527328130508919157105, −4.95152481977458114048720514109, −3.60126515952798956372557416511, −2.41293762330625200421368580268, −1.52265716070339723185493718732, 0,
1.52265716070339723185493718732, 2.41293762330625200421368580268, 3.60126515952798956372557416511, 4.95152481977458114048720514109, 5.63901990527328130508919157105, 6.33716156265809256001812396913, 7.15924808448895041647147210291, 7.80649049598333949380109845076, 9.084375042641264038456207040056
