| L(s) = 1 | + 3-s − 4·5-s + 2·7-s + 9-s + 4·11-s + 13-s − 4·15-s + 2·17-s + 2·19-s + 2·21-s + 11·25-s + 27-s − 6·29-s + 10·31-s + 4·33-s − 8·35-s + 10·37-s + 39-s + 8·41-s − 4·43-s − 4·45-s + 4·47-s − 3·49-s + 2·51-s − 10·53-s − 16·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 1.03·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 11/5·25-s + 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.696·33-s − 1.35·35-s + 1.64·37-s + 0.160·39-s + 1.24·41-s − 0.609·43-s − 0.596·45-s + 0.583·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 2.15·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.582186746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582186746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.84045893410549491261609677564, −9.552899241265083891285039757219, −8.701706605624698981393407700410, −7.87838450623577993623959455577, −7.46781649615290225514077683026, −6.25490600491551353306957903185, −4.64506000416277389326922266092, −4.02508633439341202211647147289, −3.05605111090680200450117437002, −1.17745705159514838549061128898,
1.17745705159514838549061128898, 3.05605111090680200450117437002, 4.02508633439341202211647147289, 4.64506000416277389326922266092, 6.25490600491551353306957903185, 7.46781649615290225514077683026, 7.87838450623577993623959455577, 8.701706605624698981393407700410, 9.552899241265083891285039757219, 10.84045893410549491261609677564
