| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 2·11-s − 2·12-s + 2·13-s − 14-s + 16-s + 17-s + 18-s + 2·21-s + 2·22-s − 4·23-s − 2·24-s − 5·25-s + 2·26-s + 4·27-s − 28-s − 4·29-s + 32-s − 4·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.577·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.436·21-s + 0.426·22-s − 0.834·23-s − 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.188·28-s − 0.742·29-s + 0.176·32-s − 0.696·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228718 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228718 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.613087620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.613087620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.91945939317058, −12.36829249493200, −11.92529823505538, −11.72299249268471, −11.11990641742877, −10.88834636128862, −10.16402499286761, −9.890597006235636, −9.336766391982855, −8.496840316243936, −8.380687675654509, −7.384512638759039, −7.194057471979432, −6.388837639175732, −6.226140643950853, −5.767561685430792, −5.213686628560588, −4.878223545760939, −4.005637580207768, −3.764185992282325, −3.259262375747695, −2.368270380474783, −1.824780365469311, −1.126363561595363, −0.3556375913513599,
0.3556375913513599, 1.126363561595363, 1.824780365469311, 2.368270380474783, 3.259262375747695, 3.764185992282325, 4.005637580207768, 4.878223545760939, 5.213686628560588, 5.767561685430792, 6.226140643950853, 6.388837639175732, 7.194057471979432, 7.384512638759039, 8.380687675654509, 8.496840316243936, 9.336766391982855, 9.890597006235636, 10.16402499286761, 10.88834636128862, 11.11990641742877, 11.72299249268471, 11.92529823505538, 12.36829249493200, 12.91945939317058
