| L(s) = 1 | − 1.23·2-s − 0.475·4-s + 2.26·5-s − 3.95·7-s + 3.05·8-s − 2.80·10-s − 11-s − 0.851·13-s + 4.87·14-s − 2.82·16-s + 3.93·17-s − 4.46·19-s − 1.07·20-s + 1.23·22-s + 6.51·23-s + 0.147·25-s + 1.05·26-s + 1.87·28-s + 3.86·29-s + 4.18·31-s − 2.62·32-s − 4.86·34-s − 8.96·35-s − 7.67·37-s + 5.51·38-s + 6.93·40-s − 5.87·41-s + ⋯ |
| L(s) = 1 | − 0.873·2-s − 0.237·4-s + 1.01·5-s − 1.49·7-s + 1.08·8-s − 0.885·10-s − 0.301·11-s − 0.236·13-s + 1.30·14-s − 0.705·16-s + 0.955·17-s − 1.02·19-s − 0.241·20-s + 0.263·22-s + 1.35·23-s + 0.0295·25-s + 0.206·26-s + 0.354·28-s + 0.717·29-s + 0.751·31-s − 0.464·32-s − 0.834·34-s − 1.51·35-s − 1.26·37-s + 0.894·38-s + 1.09·40-s − 0.918·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 13 | \( 1 + 0.851T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 - 3.86T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 7.67T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 3.99T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 + 5.28T + 73T^{2} \) |
| 79 | \( 1 - 8.65T + 79T^{2} \) |
| 83 | \( 1 + 4.26T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 + 4.41T + 97T^{2} \) |
| 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
−7.48410865515067373502256500304, −6.79903031650786038213523626163, −6.30999032065312436461364991705, −5.40822635250652331843344686558, −4.85418991599941216697679243495, −3.74711531143678812475040602253, −2.99215773989237313741964744803, −2.09812488907444361618394878042, −1.06000767511680617662607239991, 0,
1.06000767511680617662607239991, 2.09812488907444361618394878042, 2.99215773989237313741964744803, 3.74711531143678812475040602253, 4.85418991599941216697679243495, 5.40822635250652331843344686558, 6.30999032065312436461364991705, 6.79903031650786038213523626163, 7.48410865515067373502256500304
