| L(s) = 1 | + 1.88·2-s + 1.88·3-s + 1.55·4-s + 3.56·6-s + 0.335·7-s − 0.836·8-s + 0.569·9-s − 3.80·11-s + 2.94·12-s + 0.558·13-s + 0.632·14-s − 4.69·16-s + 1.07·18-s − 3.73·19-s + 0.633·21-s − 7.18·22-s + 1.24·23-s − 1.58·24-s + 1.05·26-s − 4.59·27-s + 0.521·28-s − 3.98·29-s − 9.36·31-s − 7.17·32-s − 7.19·33-s + 0.887·36-s + 6.77·37-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 1.09·3-s + 0.778·4-s + 1.45·6-s + 0.126·7-s − 0.295·8-s + 0.189·9-s − 1.14·11-s + 0.848·12-s + 0.154·13-s + 0.169·14-s − 1.17·16-s + 0.253·18-s − 0.856·19-s + 0.138·21-s − 1.53·22-s + 0.259·23-s − 0.322·24-s + 0.206·26-s − 0.883·27-s + 0.0986·28-s − 0.740·29-s − 1.68·31-s − 1.26·32-s − 1.25·33-s + 0.147·36-s + 1.11·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.88T + 2T^{2} \) |
| 3 | \( 1 - 1.88T + 3T^{2} \) |
| 7 | \( 1 - 0.335T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 - 0.558T + 13T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 3.98T + 29T^{2} \) |
| 31 | \( 1 + 9.36T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 - 9.50T + 47T^{2} \) |
| 53 | \( 1 - 7.30T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 0.743T + 61T^{2} \) |
| 67 | \( 1 + 7.29T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64027052650438020094685611538, −6.81324709434217771930770458624, −5.86644428194017450752063374726, −5.44748645496610974723677577735, −4.58275986487422143192392608800, −3.91124466051000483294378648680, −3.21615478631743197588691421302, −2.57422302469991083216391374153, −1.89924217558148589626514521696, 0,
1.89924217558148589626514521696, 2.57422302469991083216391374153, 3.21615478631743197588691421302, 3.91124466051000483294378648680, 4.58275986487422143192392608800, 5.44748645496610974723677577735, 5.86644428194017450752063374726, 6.81324709434217771930770458624, 7.64027052650438020094685611538
