| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.209020544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209020544\) |
| \(L(\frac{3}{2})\) |
|
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
−9.042400737747083320680488489846, −8.465783800051715901501638907774, −7.05413896124171905710563427400, −6.74871626085048175906066789560, −5.79370439967915340874746227234, −5.21398008013466937295056850606, −3.92416873404758318228720658701, −3.06691422745968021537813590492, −2.12759928249699131341024201812, −1.23681299563997649651316036995,
1.23681299563997649651316036995, 2.12759928249699131341024201812, 3.06691422745968021537813590492, 3.92416873404758318228720658701, 5.21398008013466937295056850606, 5.79370439967915340874746227234, 6.74871626085048175906066789560, 7.05413896124171905710563427400, 8.465783800051715901501638907774, 9.042400737747083320680488489846
