| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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
−8.233189527945701230680574967528, −7.34625463998216708225115176322, −6.75025046582949378189545643050, −5.87026022849043240716165359829, −5.15247336607215317330370575493, −4.15462631442213561591336148099, −3.52709486463188557989681977937, −2.19444978242734083122256032300, −1.10576222542035645039706575422, 0,
1.10576222542035645039706575422, 2.19444978242734083122256032300, 3.52709486463188557989681977937, 4.15462631442213561591336148099, 5.15247336607215317330370575493, 5.87026022849043240716165359829, 6.75025046582949378189545643050, 7.34625463998216708225115176322, 8.233189527945701230680574967528
