| L(s) = 1 | + 2.77·2-s + 5.71·4-s + 2.71·7-s + 10.3·8-s + 2.71·11-s − 13-s + 7.55·14-s + 17.2·16-s − 2.83·17-s − 3.55·19-s + 7.55·22-s − 4.83·23-s − 2.77·26-s + 15.5·28-s − 6·29-s + 7.55·31-s + 27.3·32-s − 7.88·34-s + 4.27·37-s − 9.88·38-s − 2.83·41-s − 11.1·43-s + 15.5·44-s − 13.4·46-s − 11.5·47-s + 0.397·49-s − 5.71·52-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 2.85·4-s + 1.02·7-s + 3.65·8-s + 0.820·11-s − 0.277·13-s + 2.01·14-s + 4.31·16-s − 0.688·17-s − 0.816·19-s + 1.61·22-s − 1.00·23-s − 0.544·26-s + 2.93·28-s − 1.11·29-s + 1.35·31-s + 4.83·32-s − 1.35·34-s + 0.703·37-s − 1.60·38-s − 0.443·41-s − 1.69·43-s + 2.34·44-s − 1.98·46-s − 1.68·47-s + 0.0567·49-s − 0.793·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.791492164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.791492164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 7 | \( 1 - 2.71T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 17 | \( 1 + 2.83T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7.55T + 31T^{2} \) |
| 37 | \( 1 - 4.27T + 37T^{2} \) |
| 41 | \( 1 + 2.83T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 1.16T + 53T^{2} \) |
| 59 | \( 1 - 2.11T + 59T^{2} \) |
| 61 | \( 1 - 6.60T + 61T^{2} \) |
| 67 | \( 1 + 1.88T + 67T^{2} \) |
| 71 | \( 1 - 6.71T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.367795974814934724227391725841, −7.85327693160653701767984681969, −6.77106278530480714808745413571, −6.42312271595294308544693808468, −5.48878774688548656088763411966, −4.71019991458121680961049209359, −4.23434181446291823830574039196, −3.39779165681467514093506649168, −2.24193553724171030200428113629, −1.61846586243664433483200706371,
1.61846586243664433483200706371, 2.24193553724171030200428113629, 3.39779165681467514093506649168, 4.23434181446291823830574039196, 4.71019991458121680961049209359, 5.48878774688548656088763411966, 6.42312271595294308544693808468, 6.77106278530480714808745413571, 7.85327693160653701767984681969, 8.367795974814934724227391725841
