| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 4·11-s − 6·13-s − 14-s + 16-s − 2·17-s + 2·20-s + 4·22-s − 8·23-s − 25-s − 6·26-s − 28-s − 2·29-s + 32-s − 2·34-s − 2·35-s + 10·37-s + 2·40-s − 6·41-s − 4·43-s + 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.342·34-s − 0.338·35-s + 1.64·37-s + 0.316·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.682674090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.682674090\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.46693955898140, −14.18319892760150, −13.62416033898230, −13.15853571183154, −12.63390415410471, −11.98493677118942, −11.79375579656525, −11.16699364739058, −10.30449464140171, −9.881943953560530, −9.593865608947072, −9.018304895975381, −8.192292784700145, −7.625851762095147, −6.912227842020914, −6.515881105093448, −5.996141961480742, −5.435209209468500, −4.814742947537908, −4.127977881209739, −3.704879412469944, −2.764228955081988, −2.155840740110551, −1.762967993837964, −0.5717430969273195,
0.5717430969273195, 1.762967993837964, 2.155840740110551, 2.764228955081988, 3.704879412469944, 4.127977881209739, 4.814742947537908, 5.435209209468500, 5.996141961480742, 6.515881105093448, 6.912227842020914, 7.625851762095147, 8.192292784700145, 9.018304895975381, 9.593865608947072, 9.881943953560530, 10.30449464140171, 11.16699364739058, 11.79375579656525, 11.98493677118942, 12.63390415410471, 13.15853571183154, 13.62416033898230, 14.18319892760150, 14.46693955898140
