| L(s) = 1 | + 2.15·2-s + 2.64·4-s + 2.62·7-s + 1.38·8-s + 1.64·11-s − 2.44·13-s + 5.65·14-s − 2.29·16-s − 0.578·17-s + 5.20·19-s + 3.54·22-s + 8.22·23-s − 5.26·26-s + 6.94·28-s − 0.766·29-s + 4.21·31-s − 7.72·32-s − 1.24·34-s − 37-s + 11.2·38-s + 1.64·41-s + 1.91·43-s + 4.34·44-s + 17.7·46-s − 9.56·47-s − 0.108·49-s − 6.45·52-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.32·4-s + 0.992·7-s + 0.491·8-s + 0.495·11-s − 0.677·13-s + 1.51·14-s − 0.573·16-s − 0.140·17-s + 1.19·19-s + 0.755·22-s + 1.71·23-s − 1.03·26-s + 1.31·28-s − 0.142·29-s + 0.756·31-s − 1.36·32-s − 0.213·34-s − 0.164·37-s + 1.81·38-s + 0.256·41-s + 0.292·43-s + 0.655·44-s + 2.61·46-s − 1.39·47-s − 0.0154·49-s − 0.895·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.009458233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.009458233\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.15T + 2T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + 0.578T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 - 8.22T + 23T^{2} \) |
| 29 | \( 1 + 0.766T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 - 1.91T + 43T^{2} \) |
| 47 | \( 1 + 9.56T + 47T^{2} \) |
| 53 | \( 1 - 7.74T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 8.85T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.50540653493635914104743392325, −6.98587555077356977125456249791, −6.29313816945331035361857204896, −5.37908055396107562653079170416, −4.99243525171362340309788111627, −4.46835893176211625848752172232, −3.57290966606442043755487328503, −2.91016553652674048904707544276, −2.06156196066286758826559176027, −0.998393840323530490448793949446,
0.998393840323530490448793949446, 2.06156196066286758826559176027, 2.91016553652674048904707544276, 3.57290966606442043755487328503, 4.46835893176211625848752172232, 4.99243525171362340309788111627, 5.37908055396107562653079170416, 6.29313816945331035361857204896, 6.98587555077356977125456249791, 7.50540653493635914104743392325
