| L(s) = 1 | + 3·3-s − 4·5-s + 9·9-s − 26·11-s − 2·13-s − 12·15-s + 36·17-s + 76·19-s − 114·23-s − 109·25-s + 27·27-s + 6·29-s + 256·31-s − 78·33-s − 86·37-s − 6·39-s − 160·41-s − 220·43-s − 36·45-s − 308·47-s + 108·51-s + 258·53-s + 104·55-s + 228·57-s − 264·59-s − 606·61-s + 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.357·5-s + 1/3·9-s − 0.712·11-s − 0.0426·13-s − 0.206·15-s + 0.513·17-s + 0.917·19-s − 1.03·23-s − 0.871·25-s + 0.192·27-s + 0.0384·29-s + 1.48·31-s − 0.411·33-s − 0.382·37-s − 0.0246·39-s − 0.609·41-s − 0.780·43-s − 0.119·45-s − 0.955·47-s + 0.296·51-s + 0.668·53-s + 0.254·55-s + 0.529·57-s − 0.582·59-s − 1.27·61-s + 0.0152·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 86 T + p^{3} T^{2} \) |
| 41 | \( 1 + 160 T + p^{3} T^{2} \) |
| 43 | \( 1 + 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 308 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 264 T + p^{3} T^{2} \) |
| 61 | \( 1 + 606 T + p^{3} T^{2} \) |
| 67 | \( 1 + 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 286 T + p^{3} T^{2} \) |
| 73 | \( 1 - 530 T + p^{3} T^{2} \) |
| 79 | \( 1 + 44 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1012 T + p^{3} T^{2} \) |
| 89 | \( 1 + 768 T + p^{3} T^{2} \) |
| 97 | \( 1 + 222 T + p^{3} T^{2} \) |
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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
−8.914122625621728135152979452247, −7.985935049542614220223016089156, −7.67155547114124852797961645769, −6.55167796795174463245946603328, −5.53772276431970753944244184792, −4.58451289667410245132225901934, −3.56065907456020323641661676846, −2.72167183679391024876990892705, −1.48998420191567421205438285669, 0,
1.48998420191567421205438285669, 2.72167183679391024876990892705, 3.56065907456020323641661676846, 4.58451289667410245132225901934, 5.53772276431970753944244184792, 6.55167796795174463245946603328, 7.67155547114124852797961645769, 7.985935049542614220223016089156, 8.914122625621728135152979452247
