| L(s) = 1 | + 2.47·2-s + 3-s + 4.12·4-s − 0.798·5-s + 2.47·6-s + 5.24·8-s + 9-s − 1.97·10-s + 0.583·11-s + 4.12·12-s − 1.18·13-s − 0.798·15-s + 4.74·16-s − 6.24·17-s + 2.47·18-s + 7.64·19-s − 3.29·20-s + 1.44·22-s + 8.22·23-s + 5.24·24-s − 4.36·25-s − 2.93·26-s + 27-s + 1.77·29-s − 1.97·30-s + 2.75·31-s + 1.23·32-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.577·3-s + 2.06·4-s − 0.357·5-s + 1.01·6-s + 1.85·8-s + 0.333·9-s − 0.624·10-s + 0.176·11-s + 1.18·12-s − 0.329·13-s − 0.206·15-s + 1.18·16-s − 1.51·17-s + 0.583·18-s + 1.75·19-s − 0.735·20-s + 0.308·22-s + 1.71·23-s + 1.07·24-s − 0.872·25-s − 0.576·26-s + 0.192·27-s + 0.329·29-s − 0.360·30-s + 0.493·31-s + 0.218·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8673 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.899063936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.899063936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 + 0.798T + 5T^{2} \) |
| 11 | \( 1 - 0.583T + 11T^{2} \) |
| 13 | \( 1 + 1.18T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 - 8.22T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 - 3.29T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 61 | \( 1 - 0.324T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−7.43680964702913854915880765533, −7.02001328950314079276449232924, −6.29976927445228791232043640332, −5.51304173281953127008583211291, −4.80990314369888449905638143392, −4.26907611296444675749162200244, −3.59509064314919583232519750915, −2.79485973511021416095785572320, −2.34751450916657853860347501158, −1.05706156684583935243795165141,
1.05706156684583935243795165141, 2.34751450916657853860347501158, 2.79485973511021416095785572320, 3.59509064314919583232519750915, 4.26907611296444675749162200244, 4.80990314369888449905638143392, 5.51304173281953127008583211291, 6.29976927445228791232043640332, 7.02001328950314079276449232924, 7.43680964702913854915880765533
