| L(s) = 1 | − 2.74·2-s + 2.49·3-s + 5.53·4-s + 0.133·5-s − 6.85·6-s + 0.778·7-s − 9.70·8-s + 3.24·9-s − 0.367·10-s + 5.21·11-s + 13.8·12-s − 3.44·13-s − 2.13·14-s + 0.334·15-s + 15.5·16-s − 3.95·17-s − 8.90·18-s + 2.36·19-s + 0.741·20-s + 1.94·21-s − 14.3·22-s − 9.52·23-s − 24.2·24-s − 4.98·25-s + 9.44·26-s + 0.611·27-s + 4.30·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.44·3-s + 2.76·4-s + 0.0599·5-s − 2.80·6-s + 0.294·7-s − 3.43·8-s + 1.08·9-s − 0.116·10-s + 1.57·11-s + 3.99·12-s − 0.954·13-s − 0.570·14-s + 0.0864·15-s + 3.89·16-s − 0.959·17-s − 2.09·18-s + 0.541·19-s + 0.165·20-s + 0.424·21-s − 3.05·22-s − 1.98·23-s − 4.95·24-s − 0.996·25-s + 1.85·26-s + 0.117·27-s + 0.813·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 2.49T + 3T^{2} \) |
| 5 | \( 1 - 0.133T + 5T^{2} \) |
| 7 | \( 1 - 0.778T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 - 2.36T + 19T^{2} \) |
| 23 | \( 1 + 9.52T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 + 0.680T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 + 7.71T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 3.41T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 1.61T + 97T^{2} \) |
| 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
−7.82513271068397812490646405440, −7.07464302786286632454109931554, −6.62979703246081258732710829413, −5.80474067094475720530117546215, −4.36318654712403683535413330762, −3.54758019612130528183714848103, −2.66849441519631176144276196263, −1.96867214481092256737230201754, −1.46019980926874041238016480512, 0,
1.46019980926874041238016480512, 1.96867214481092256737230201754, 2.66849441519631176144276196263, 3.54758019612130528183714848103, 4.36318654712403683535413330762, 5.80474067094475720530117546215, 6.62979703246081258732710829413, 7.07464302786286632454109931554, 7.82513271068397812490646405440
