| L(s) = 1 | + 3·2-s + 4·3-s + 4-s + 12·6-s − 28·7-s − 21·8-s − 11·9-s + 2·11-s + 4·12-s − 13·13-s − 84·14-s − 71·16-s + 44·17-s − 33·18-s − 94·19-s − 112·21-s + 6·22-s − 18·23-s − 84·24-s − 39·26-s − 152·27-s − 28·28-s + 118·29-s − 100·31-s − 45·32-s + 8·33-s + 132·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.769·3-s + 1/8·4-s + 0.816·6-s − 1.51·7-s − 0.928·8-s − 0.407·9-s + 0.0548·11-s + 0.0962·12-s − 0.277·13-s − 1.60·14-s − 1.10·16-s + 0.627·17-s − 0.432·18-s − 1.13·19-s − 1.16·21-s + 0.0581·22-s − 0.163·23-s − 0.714·24-s − 0.294·26-s − 1.08·27-s − 0.188·28-s + 0.755·29-s − 0.579·31-s − 0.248·32-s + 0.0422·33-s + 0.665·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 94 T + p^{3} T^{2} \) |
| 23 | \( 1 + 18 T + p^{3} T^{2} \) |
| 29 | \( 1 - 118 T + p^{3} T^{2} \) |
| 31 | \( 1 + 100 T + p^{3} T^{2} \) |
| 37 | \( 1 - 126 T + p^{3} T^{2} \) |
| 41 | \( 1 - 474 T + p^{3} T^{2} \) |
| 43 | \( 1 + 200 T + p^{3} T^{2} \) |
| 47 | \( 1 - 448 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 446 T + p^{3} T^{2} \) |
| 61 | \( 1 + 638 T + p^{3} T^{2} \) |
| 67 | \( 1 + 868 T + p^{3} T^{2} \) |
| 71 | \( 1 - 536 T + p^{3} T^{2} \) |
| 73 | \( 1 + 58 T + p^{3} T^{2} \) |
| 79 | \( 1 - 232 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 774 T + p^{3} T^{2} \) |
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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
−10.75092930939693284723606093920, −9.520519027989233432680990132767, −9.042579485645581565225927232231, −7.83989831363987479067694845007, −6.45668347978606200688252498902, −5.79411317698717531994010671114, −4.36117034283159561696631141383, −3.34947305612004350561343049379, −2.62438468937431018270592078084, 0,
2.62438468937431018270592078084, 3.34947305612004350561343049379, 4.36117034283159561696631141383, 5.79411317698717531994010671114, 6.45668347978606200688252498902, 7.83989831363987479067694845007, 9.042579485645581565225927232231, 9.520519027989233432680990132767, 10.75092930939693284723606093920
