| L(s) = 1 | − 1.35·2-s + 0.875·3-s − 0.176·4-s + 5-s − 1.18·6-s + 2.93·8-s − 2.23·9-s − 1.35·10-s + 4.84·11-s − 0.154·12-s + 0.579·13-s + 0.875·15-s − 3.61·16-s + 1.03·17-s + 3.01·18-s + 3.81·19-s − 0.176·20-s − 6.54·22-s − 5.72·23-s + 2.57·24-s + 25-s − 0.782·26-s − 4.58·27-s − 5.14·29-s − 1.18·30-s + 31-s − 0.994·32-s + ⋯ |
| L(s) = 1 | − 0.954·2-s + 0.505·3-s − 0.0881·4-s + 0.447·5-s − 0.482·6-s + 1.03·8-s − 0.744·9-s − 0.427·10-s + 1.46·11-s − 0.0445·12-s + 0.160·13-s + 0.226·15-s − 0.904·16-s + 0.249·17-s + 0.710·18-s + 0.874·19-s − 0.0394·20-s − 1.39·22-s − 1.19·23-s + 0.525·24-s + 0.200·25-s − 0.153·26-s − 0.882·27-s − 0.956·29-s − 0.215·30-s + 0.179·31-s − 0.175·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472484754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472484754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 - 0.875T + 3T^{2} \) |
| 11 | \( 1 - 4.84T + 11T^{2} \) |
| 13 | \( 1 - 0.579T + 13T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 - 3.81T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 + 3.02T + 43T^{2} \) |
| 47 | \( 1 + 0.883T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 - 0.609T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 0.949T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8.28T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 4.23T + 89T^{2} \) |
| 97 | \( 1 + 7.74T + 97T^{2} \) |
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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
−8.081945432213824674599144038127, −7.42472504170715680897516960320, −6.62462703352481069288687128193, −5.86419857543424535894651568021, −5.17727870762681165114048806039, −4.07090118504013025443203774765, −3.61216557440748951789586850439, −2.44980365745811488657290123399, −1.62874063313166207994855569349, −0.72213164751456000575957538268,
0.72213164751456000575957538268, 1.62874063313166207994855569349, 2.44980365745811488657290123399, 3.61216557440748951789586850439, 4.07090118504013025443203774765, 5.17727870762681165114048806039, 5.86419857543424535894651568021, 6.62462703352481069288687128193, 7.42472504170715680897516960320, 8.081945432213824674599144038127
