| L(s) = 1 | − 0.0935·2-s − 1.46·3-s − 1.99·4-s + 0.137·6-s − 4.52·7-s + 0.373·8-s − 0.848·9-s + 2.92·12-s + 1.14·13-s + 0.423·14-s + 3.94·16-s − 3.37·17-s + 0.0793·18-s + 6.08·19-s + 6.63·21-s + 5.45·23-s − 0.547·24-s − 0.106·26-s + 5.64·27-s + 9.00·28-s + 3.32·29-s + 1.79·31-s − 1.11·32-s + 0.315·34-s + 1.68·36-s − 1.48·37-s − 0.568·38-s + ⋯ |
| L(s) = 1 | − 0.0661·2-s − 0.846·3-s − 0.995·4-s + 0.0559·6-s − 1.71·7-s + 0.131·8-s − 0.282·9-s + 0.843·12-s + 0.316·13-s + 0.113·14-s + 0.986·16-s − 0.818·17-s + 0.0187·18-s + 1.39·19-s + 1.44·21-s + 1.13·23-s − 0.111·24-s − 0.0209·26-s + 1.08·27-s + 1.70·28-s + 0.616·29-s + 0.321·31-s − 0.197·32-s + 0.0540·34-s + 0.281·36-s − 0.244·37-s − 0.0923·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.0935T + 2T^{2} \) |
| 3 | \( 1 + 1.46T + 3T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 1.79T + 31T^{2} \) |
| 37 | \( 1 + 1.48T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 43 | \( 1 - 0.263T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 - 2.50T + 61T^{2} \) |
| 67 | \( 1 - 0.516T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.68T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.35T + 97T^{2} \) |
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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
−8.663381433167201349977024622560, −7.48685762320924094786381458579, −6.64034510553072870855756965958, −6.07450536497259733844390902359, −5.29182114688552364639002774251, −4.60115969118507816059526945193, −3.46896006353525058391645238817, −2.92360927880467552357364148719, −0.967911907256866605583460300236, 0,
0.967911907256866605583460300236, 2.92360927880467552357364148719, 3.46896006353525058391645238817, 4.60115969118507816059526945193, 5.29182114688552364639002774251, 6.07450536497259733844390902359, 6.64034510553072870855756965958, 7.48685762320924094786381458579, 8.663381433167201349977024622560
