| L(s) = 1 | − 26·3-s − 16·5-s + 433·9-s + 8·11-s − 684·13-s + 416·15-s + 2.21e3·17-s + 2.69e3·19-s + 3.34e3·23-s − 2.86e3·25-s − 4.94e3·27-s − 3.25e3·29-s − 4.78e3·31-s − 208·33-s − 1.14e4·37-s + 1.77e4·39-s − 1.33e4·41-s − 928·43-s − 6.92e3·45-s − 1.21e3·47-s − 5.76e4·51-s + 1.31e4·53-s − 128·55-s − 7.01e4·57-s − 3.47e4·59-s + 1.03e3·61-s + 1.09e4·65-s + ⋯ |
| L(s) = 1 | − 1.66·3-s − 0.286·5-s + 1.78·9-s + 0.0199·11-s − 1.12·13-s + 0.477·15-s + 1.86·17-s + 1.71·19-s + 1.31·23-s − 0.918·25-s − 1.30·27-s − 0.718·29-s − 0.894·31-s − 0.0332·33-s − 1.37·37-s + 1.87·39-s − 1.24·41-s − 0.0765·43-s − 0.510·45-s − 0.0800·47-s − 3.10·51-s + 0.641·53-s − 0.00570·55-s − 2.85·57-s − 1.29·59-s + 0.0355·61-s + 0.321·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 26 T + p^{5} T^{2} \) |
| 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 11 | \( 1 - 8 T + p^{5} T^{2} \) |
| 13 | \( 1 + 684 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 19 | \( 1 - 142 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 3344 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3254 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4788 T + p^{5} T^{2} \) |
| 37 | \( 1 + 310 p T + p^{5} T^{2} \) |
| 41 | \( 1 + 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 + 928 T + p^{5} T^{2} \) |
| 47 | \( 1 + 1212 T + p^{5} T^{2} \) |
| 53 | \( 1 - 13110 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34702 T + p^{5} T^{2} \) |
| 61 | \( 1 - 1032 T + p^{5} T^{2} \) |
| 67 | \( 1 - 10108 T + p^{5} T^{2} \) |
| 71 | \( 1 - 62720 T + p^{5} T^{2} \) |
| 73 | \( 1 - 18926 T + p^{5} T^{2} \) |
| 79 | \( 1 - 11400 T + p^{5} T^{2} \) |
| 83 | \( 1 + 88958 T + p^{5} T^{2} \) |
| 89 | \( 1 + 19722 T + p^{5} T^{2} \) |
| 97 | \( 1 + 17062 T + p^{5} T^{2} \) |
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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
−11.33471956710895150907029557543, −10.25188740449509304852350380857, −9.509433783961479908174097382477, −7.65460118025772319897637611203, −6.98565635072543990461380114714, −5.48558806131182120012246832133, −5.14065714893607491960918807244, −3.45912737019757210542685232567, −1.26148521739396053398035059599, 0,
1.26148521739396053398035059599, 3.45912737019757210542685232567, 5.14065714893607491960918807244, 5.48558806131182120012246832133, 6.98565635072543990461380114714, 7.65460118025772319897637611203, 9.509433783961479908174097382477, 10.25188740449509304852350380857, 11.33471956710895150907029557543
