| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 4·13-s + 14-s − 16-s + 2·17-s − 4·26-s − 28-s − 8·29-s − 4·31-s + 5·32-s + 2·34-s − 8·37-s − 4·41-s − 8·43-s − 12·47-s + 49-s + 4·52-s + 6·53-s − 3·56-s − 8·58-s − 8·59-s + 10·61-s − 4·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.784·26-s − 0.188·28-s − 1.48·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s − 1.31·37-s − 0.624·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s − 0.400·56-s − 1.05·58-s − 1.04·59-s + 1.28·61-s − 0.508·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.088607817903251031326638341111, −8.228551719968199269777145259992, −7.41101673575943033942193794264, −6.49064642044409101247116318141, −5.31492127060484072085833685842, −5.07400726366767548775789169629, −3.94638205271297856522955156636, −3.16684350678120451907018121851, −1.85825874226295916819253260244, 0,
1.85825874226295916819253260244, 3.16684350678120451907018121851, 3.94638205271297856522955156636, 5.07400726366767548775789169629, 5.31492127060484072085833685842, 6.49064642044409101247116318141, 7.41101673575943033942193794264, 8.228551719968199269777145259992, 9.088607817903251031326638341111
