| L(s) = 1 | − 3-s + 2·7-s + 9-s + 3·11-s + 13-s + 2·17-s + 8·19-s − 2·21-s + 3·23-s − 27-s + 8·29-s − 9·31-s − 3·33-s + 10·37-s − 39-s − 3·41-s + 43-s − 7·47-s − 3·49-s − 2·51-s + 5·53-s − 8·57-s + 9·59-s + 2·61-s + 2·63-s − 8·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.485·17-s + 1.83·19-s − 0.436·21-s + 0.625·23-s − 0.192·27-s + 1.48·29-s − 1.61·31-s − 0.522·33-s + 1.64·37-s − 0.160·39-s − 0.468·41-s + 0.152·43-s − 1.02·47-s − 3/7·49-s − 0.280·51-s + 0.686·53-s − 1.05·57-s + 1.17·59-s + 0.256·61-s + 0.251·63-s − 0.977·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.187600337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.187600337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.54757360244970, −14.02808194177407, −13.48281702640973, −12.94813408494133, −12.28075284914447, −11.85026830708244, −11.39155377786929, −11.11533066525806, −10.39486049427333, −9.777343977577921, −9.400966399061466, −8.765987996467928, −8.142934862778672, −7.580970204039859, −7.097370208790405, −6.483595227516096, −5.906995171543325, −5.225946287230390, −4.930436995238092, −4.164792672980344, −3.520595596177711, −2.926193766140677, −1.919222485111997, −1.192205976769917, −0.7696137595730807,
0.7696137595730807, 1.192205976769917, 1.919222485111997, 2.926193766140677, 3.520595596177711, 4.164792672980344, 4.930436995238092, 5.225946287230390, 5.906995171543325, 6.483595227516096, 7.097370208790405, 7.580970204039859, 8.142934862778672, 8.765987996467928, 9.400966399061466, 9.777343977577921, 10.39486049427333, 11.11533066525806, 11.39155377786929, 11.85026830708244, 12.28075284914447, 12.94813408494133, 13.48281702640973, 14.02808194177407, 14.54757360244970
