| L(s) = 1 | + 2-s + 0.673·3-s + 4-s − 0.182·5-s + 0.673·6-s − 1.88·7-s + 8-s − 2.54·9-s − 0.182·10-s − 2.50·11-s + 0.673·12-s + 3.59·13-s − 1.88·14-s − 0.123·15-s + 16-s + 5.20·17-s − 2.54·18-s − 3.03·19-s − 0.182·20-s − 1.26·21-s − 2.50·22-s − 1.52·23-s + 0.673·24-s − 4.96·25-s + 3.59·26-s − 3.73·27-s − 1.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.388·3-s + 0.5·4-s − 0.0818·5-s + 0.274·6-s − 0.711·7-s + 0.353·8-s − 0.848·9-s − 0.0578·10-s − 0.756·11-s + 0.194·12-s + 0.996·13-s − 0.503·14-s − 0.0318·15-s + 0.250·16-s + 1.26·17-s − 0.600·18-s − 0.695·19-s − 0.0409·20-s − 0.276·21-s − 0.535·22-s − 0.318·23-s + 0.137·24-s − 0.993·25-s + 0.704·26-s − 0.718·27-s − 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 0.673T + 3T^{2} \) |
| 5 | \( 1 + 0.182T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 - 5.20T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 - 0.922T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 + 8.70T + 53T^{2} \) |
| 59 | \( 1 - 9.56T + 59T^{2} \) |
| 61 | \( 1 - 8.41T + 61T^{2} \) |
| 67 | \( 1 + 5.72T + 67T^{2} \) |
| 71 | \( 1 + 1.24T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 0.365T + 97T^{2} \) |
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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
−8.036400189994382436461621046316, −7.39453905693269329351840528641, −6.38624937254964043377076666592, −5.77570994165205943055354067177, −5.27517276753153910100838360874, −3.98155734161885673087557678618, −3.46396612005658726690010630611, −2.74080713562106880038760117920, −1.71013606928752884148911452490, 0,
1.71013606928752884148911452490, 2.74080713562106880038760117920, 3.46396612005658726690010630611, 3.98155734161885673087557678618, 5.27517276753153910100838360874, 5.77570994165205943055354067177, 6.38624937254964043377076666592, 7.39453905693269329351840528641, 8.036400189994382436461621046316
