| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 5-s − 7-s + 2.23·8-s + 0.618·10-s + 3.23·11-s + 2.47·13-s + 0.618·14-s + 1.85·16-s − 4.23·17-s − 7.47·19-s + 1.61·20-s − 2.00·22-s + 6.23·23-s + 25-s − 1.52·26-s + 1.61·28-s + 7.70·29-s − 9·31-s − 5.61·32-s + 2.61·34-s + 35-s − 9.70·37-s + 4.61·38-s − 2.23·40-s − 8.47·41-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 0.447·5-s − 0.377·7-s + 0.790·8-s + 0.195·10-s + 0.975·11-s + 0.685·13-s + 0.165·14-s + 0.463·16-s − 1.02·17-s − 1.71·19-s + 0.361·20-s − 0.426·22-s + 1.30·23-s + 0.200·25-s − 0.299·26-s + 0.305·28-s + 1.43·29-s − 1.61·31-s − 0.993·32-s + 0.448·34-s + 0.169·35-s − 1.59·37-s + 0.749·38-s − 0.353·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 17 | \( 1 + 4.23T + 17T^{2} \) |
| 19 | \( 1 + 7.47T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 0.708T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 0.472T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 11T + 83T^{2} \) |
| 89 | \( 1 - 8.18T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−9.391706622678198002854208922000, −8.729898990427414556258071285606, −8.376995478464307025008966230424, −6.98856901807420140047761268445, −6.45992653265138880894882843728, −5.05497557256079244629396029482, −4.20756096555383868349042482036, −3.38679108748599707894809493918, −1.59538412249758020081216197770, 0,
1.59538412249758020081216197770, 3.38679108748599707894809493918, 4.20756096555383868349042482036, 5.05497557256079244629396029482, 6.45992653265138880894882843728, 6.98856901807420140047761268445, 8.376995478464307025008966230424, 8.729898990427414556258071285606, 9.391706622678198002854208922000
