| L(s) = 1 | + 2.51·2-s − 3-s + 4.32·4-s − 5-s − 2.51·6-s + 0.485·7-s + 5.83·8-s + 9-s − 2.51·10-s + 5.02·11-s − 4.32·12-s + 3.51·13-s + 1.22·14-s + 15-s + 6.02·16-s − 1.32·17-s + 2.51·18-s − 6.64·19-s − 4.32·20-s − 0.485·21-s + 12.6·22-s + 0.292·23-s − 5.83·24-s + 25-s + 8.83·26-s − 27-s + 2.09·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.577·3-s + 2.16·4-s − 0.447·5-s − 1.02·6-s + 0.183·7-s + 2.06·8-s + 0.333·9-s − 0.795·10-s + 1.51·11-s − 1.24·12-s + 0.974·13-s + 0.326·14-s + 0.258·15-s + 1.50·16-s − 0.320·17-s + 0.592·18-s − 1.52·19-s − 0.966·20-s − 0.106·21-s + 2.69·22-s + 0.0610·23-s − 1.19·24-s + 0.200·25-s + 1.73·26-s − 0.192·27-s + 0.396·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.240552497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.240552497\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 7 | \( 1 - 0.485T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 13 | \( 1 - 3.51T + 13T^{2} \) |
| 17 | \( 1 + 1.32T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 23 | \( 1 - 0.292T + 23T^{2} \) |
| 29 | \( 1 + 9.86T + 29T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 + 1.02T + 43T^{2} \) |
| 47 | \( 1 + 6.93T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 0.349T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 5.03T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−11.27120730092607159646938595120, −10.87969063594727559620531205855, −9.306449686427642405968044345530, −8.052523585740122725467835182793, −6.59365449872797049957831125493, −6.42806372926683765533388354856, −5.19136708332108741867952233904, −4.15878588390051660152297709177, −3.61730455444591558653163868107, −1.81131322802617597152321194068,
1.81131322802617597152321194068, 3.61730455444591558653163868107, 4.15878588390051660152297709177, 5.19136708332108741867952233904, 6.42806372926683765533388354856, 6.59365449872797049957831125493, 8.052523585740122725467835182793, 9.306449686427642405968044345530, 10.87969063594727559620531205855, 11.27120730092607159646938595120
