| L(s) = 1 | − 0.753·2-s + 2.87·3-s − 1.43·4-s + 3.13·5-s − 2.16·6-s + 0.322·7-s + 2.58·8-s + 5.26·9-s − 2.36·10-s + 3.53·11-s − 4.11·12-s − 2.34·13-s − 0.243·14-s + 9.01·15-s + 0.912·16-s + 17-s − 3.97·18-s − 7.79·19-s − 4.48·20-s + 0.928·21-s − 2.66·22-s − 4.30·23-s + 7.43·24-s + 4.82·25-s + 1.76·26-s + 6.51·27-s − 0.462·28-s + ⋯ |
| L(s) = 1 | − 0.533·2-s + 1.65·3-s − 0.715·4-s + 1.40·5-s − 0.884·6-s + 0.122·7-s + 0.914·8-s + 1.75·9-s − 0.747·10-s + 1.06·11-s − 1.18·12-s − 0.649·13-s − 0.0650·14-s + 2.32·15-s + 0.228·16-s + 0.242·17-s − 0.935·18-s − 1.78·19-s − 1.00·20-s + 0.202·21-s − 0.568·22-s − 0.897·23-s + 1.51·24-s + 0.965·25-s + 0.345·26-s + 1.25·27-s − 0.0873·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.225840993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.225840993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 0.753T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 7 | \( 1 - 0.322T + 7T^{2} \) |
| 11 | \( 1 - 3.53T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 + 4.30T + 23T^{2} \) |
| 29 | \( 1 - 9.30T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 4.85T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 1.37T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 1.87T + 89T^{2} \) |
| 97 | \( 1 + 9.00T + 97T^{2} \) |
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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
−9.897967480637636521529057568812, −9.485656733299445499795041816266, −8.730207843179682447318693787736, −8.261430984414120428959751430302, −7.13897645602114850084858711918, −6.09477543003199636746311316274, −4.69537845064226096206012274373, −3.81789122196976565682558234939, −2.40163288442299338654884529300, −1.58290624371752045044603173505,
1.58290624371752045044603173505, 2.40163288442299338654884529300, 3.81789122196976565682558234939, 4.69537845064226096206012274373, 6.09477543003199636746311316274, 7.13897645602114850084858711918, 8.261430984414120428959751430302, 8.730207843179682447318693787736, 9.485656733299445499795041816266, 9.897967480637636521529057568812
