| L(s) = 1 | − 2.60·2-s + 4.76·4-s + 0.540·5-s + 4.31·7-s − 7.18·8-s − 1.40·10-s + 11-s − 6.04·13-s − 11.2·14-s + 9.16·16-s + 1.53·17-s + 5.71·19-s + 2.57·20-s − 2.60·22-s − 2.05·23-s − 4.70·25-s + 15.7·26-s + 20.5·28-s + 0.0227·29-s − 7.86·31-s − 9.46·32-s − 3.98·34-s + 2.33·35-s − 6.22·37-s − 14.8·38-s − 3.88·40-s − 7.86·41-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 2.38·4-s + 0.241·5-s + 1.63·7-s − 2.54·8-s − 0.444·10-s + 0.301·11-s − 1.67·13-s − 2.99·14-s + 2.29·16-s + 0.371·17-s + 1.31·19-s + 0.575·20-s − 0.554·22-s − 0.428·23-s − 0.941·25-s + 3.08·26-s + 3.88·28-s + 0.00421·29-s − 1.41·31-s − 1.67·32-s − 0.683·34-s + 0.394·35-s − 1.02·37-s − 2.41·38-s − 0.614·40-s − 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 - 0.540T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 0.0227T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 - 5.04T + 53T^{2} \) |
| 59 | \( 1 + 5.62T + 59T^{2} \) |
| 67 | \( 1 - 8.08T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 8.84T + 79T^{2} \) |
| 83 | \( 1 + 9.24T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 9.60T + 97T^{2} \) |
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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
−7.934056913477710927675257919855, −7.25818928139523788393897679407, −6.85176610438001056333856795312, −5.47037487621576333692848340156, −5.23485554466014382619962921732, −3.93274827145823099887324724843, −2.68196362627306643338606051678, −1.89012131649781892706712638102, −1.33719031040155797518863550923, 0,
1.33719031040155797518863550923, 1.89012131649781892706712638102, 2.68196362627306643338606051678, 3.93274827145823099887324724843, 5.23485554466014382619962921732, 5.47037487621576333692848340156, 6.85176610438001056333856795312, 7.25818928139523788393897679407, 7.934056913477710927675257919855
