| L(s) = 1 | + 3·2-s − 3·3-s + 4-s − 9·6-s − 4·7-s − 21·8-s + 9·9-s + 34·11-s − 3·12-s − 13·13-s − 12·14-s − 71·16-s + 100·17-s + 27·18-s − 22·19-s + 12·21-s + 102·22-s + 150·23-s + 63·24-s − 39·26-s − 27·27-s − 4·28-s + 14·29-s − 292·31-s − 45·32-s − 102·33-s + 300·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 1/8·4-s − 0.612·6-s − 0.215·7-s − 0.928·8-s + 1/3·9-s + 0.931·11-s − 0.0721·12-s − 0.277·13-s − 0.229·14-s − 1.10·16-s + 1.42·17-s + 0.353·18-s − 0.265·19-s + 0.124·21-s + 0.988·22-s + 1.35·23-s + 0.535·24-s − 0.294·26-s − 0.192·27-s − 0.0269·28-s + 0.0896·29-s − 1.69·31-s − 0.248·32-s − 0.538·33-s + 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 100 T + p^{3} T^{2} \) |
| 19 | \( 1 + 22 T + p^{3} T^{2} \) |
| 23 | \( 1 - 150 T + p^{3} T^{2} \) |
| 29 | \( 1 - 14 T + p^{3} T^{2} \) |
| 31 | \( 1 + 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 354 T + p^{3} T^{2} \) |
| 41 | \( 1 + 102 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 448 T + p^{3} T^{2} \) |
| 53 | \( 1 - 238 T + p^{3} T^{2} \) |
| 59 | \( 1 + 190 T + p^{3} T^{2} \) |
| 61 | \( 1 + 254 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1096 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 114 T + p^{3} T^{2} \) |
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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
−9.326361920252276622050206724378, −8.432082149770234619060488010854, −7.10800996455755606347724295855, −6.49961945897394505362210536721, −5.44928901029438411201282536659, −4.99378833865373447885879001239, −3.80964159548380296791463602906, −3.17535795787674237370313540261, −1.48478042790236285103532489268, 0,
1.48478042790236285103532489268, 3.17535795787674237370313540261, 3.80964159548380296791463602906, 4.99378833865373447885879001239, 5.44928901029438411201282536659, 6.49961945897394505362210536721, 7.10800996455755606347724295855, 8.432082149770234619060488010854, 9.326361920252276622050206724378
