| L(s) = 1 | + 0.171·5-s + 2.92·7-s − 0.171·11-s − 3.29·13-s + 3.91·17-s − 5.81·19-s + 2.35·23-s − 4.97·25-s − 8.91·29-s − 1.15·31-s + 0.501·35-s + 6.88·37-s + 6.93·41-s − 11.1·43-s − 2.22·47-s + 1.56·49-s − 10.5·53-s − 0.0293·55-s − 1.88·59-s − 9.31·61-s − 0.564·65-s + 7.04·67-s − 3.10·71-s − 11.0·73-s − 0.501·77-s − 5.12·79-s + 5.60·83-s + ⋯ |
| L(s) = 1 | + 0.0765·5-s + 1.10·7-s − 0.0516·11-s − 0.914·13-s + 0.949·17-s − 1.33·19-s + 0.491·23-s − 0.994·25-s − 1.65·29-s − 0.208·31-s + 0.0847·35-s + 1.13·37-s + 1.08·41-s − 1.70·43-s − 0.324·47-s + 0.223·49-s − 1.45·53-s − 0.00395·55-s − 0.245·59-s − 1.19·61-s − 0.0700·65-s + 0.861·67-s − 0.368·71-s − 1.29·73-s − 0.0571·77-s − 0.576·79-s + 0.614·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.171T + 5T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 + 0.171T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 + 5.81T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 8.91T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 - 6.88T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 7.04T + 67T^{2} \) |
| 71 | \( 1 + 3.10T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.85039541662294365859905392130, −7.18506239382232576271398821938, −6.21989430976634413213100900175, −5.53900944520646407163321187984, −4.80721138631324022318402012982, −4.19974539246401247614569251246, −3.20531920475735990823439026240, −2.16858762481475496912894974238, −1.50289929976183186546055254327, 0,
1.50289929976183186546055254327, 2.16858762481475496912894974238, 3.20531920475735990823439026240, 4.19974539246401247614569251246, 4.80721138631324022318402012982, 5.53900944520646407163321187984, 6.21989430976634413213100900175, 7.18506239382232576271398821938, 7.85039541662294365859905392130
