| L(s) = 1 | + 0.311·2-s − 1.90·4-s + 5-s + 2.21·7-s − 1.21·8-s + 0.311·10-s + 1.31·11-s − 1.59·13-s + 0.688·14-s + 3.42·16-s + 0.474·17-s − 19-s − 1.90·20-s + 0.407·22-s + 7.33·23-s + 25-s − 0.495·26-s − 4.21·28-s + 0.260·29-s + 3.95·31-s + 3.49·32-s + 0.147·34-s + 2.21·35-s + 0.541·37-s − 0.311·38-s − 1.21·40-s + 12.1·41-s + ⋯ |
| L(s) = 1 | + 0.219·2-s − 0.951·4-s + 0.447·5-s + 0.836·7-s − 0.429·8-s + 0.0983·10-s + 0.395·11-s − 0.441·13-s + 0.184·14-s + 0.857·16-s + 0.115·17-s − 0.229·19-s − 0.425·20-s + 0.0869·22-s + 1.52·23-s + 0.200·25-s − 0.0971·26-s − 0.796·28-s + 0.0483·29-s + 0.710·31-s + 0.617·32-s + 0.0253·34-s + 0.374·35-s + 0.0889·37-s − 0.0504·38-s − 0.192·40-s + 1.90·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671092845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671092845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 7 | \( 1 - 2.21T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 - 0.474T + 17T^{2} \) |
| 23 | \( 1 - 7.33T + 23T^{2} \) |
| 29 | \( 1 - 0.260T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 0.541T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.18T + 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 + 5.67T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−10.07124874704099339965916422019, −9.272451787603922712965778559765, −8.623142046448662802779387020131, −7.74655219938714719654150948772, −6.67045817635063358960195872604, −5.53903109360479029487685492327, −4.86826853667779457999406165054, −4.00369461453057707547905721915, −2.66414935165163845255731296671, −1.10190631297976832572592823709,
1.10190631297976832572592823709, 2.66414935165163845255731296671, 4.00369461453057707547905721915, 4.86826853667779457999406165054, 5.53903109360479029487685492327, 6.67045817635063358960195872604, 7.74655219938714719654150948772, 8.623142046448662802779387020131, 9.272451787603922712965778559765, 10.07124874704099339965916422019
