| L(s) = 1 | + 1.59·2-s + 3·3-s − 5.44·4-s + 17.2·5-s + 4.79·6-s − 21.5·8-s + 9·9-s + 27.6·10-s + 11·11-s − 16.3·12-s − 46.1·13-s + 51.8·15-s + 9.11·16-s − 19.8·17-s + 14.3·18-s − 76.5·19-s − 93.9·20-s + 17.5·22-s − 163.·23-s − 64.5·24-s + 173.·25-s − 73.7·26-s + 27·27-s + 158.·29-s + 82.9·30-s − 170.·31-s + 186.·32-s + ⋯ |
| L(s) = 1 | + 0.565·2-s + 0.577·3-s − 0.680·4-s + 1.54·5-s + 0.326·6-s − 0.950·8-s + 0.333·9-s + 0.874·10-s + 0.301·11-s − 0.392·12-s − 0.983·13-s + 0.892·15-s + 0.142·16-s − 0.283·17-s + 0.188·18-s − 0.924·19-s − 1.05·20-s + 0.170·22-s − 1.47·23-s − 0.548·24-s + 1.38·25-s − 0.556·26-s + 0.192·27-s + 1.01·29-s + 0.504·30-s − 0.989·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.59T + 8T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 13 | \( 1 + 46.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.33T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 21.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 618.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 929.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 868.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 152.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 100.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.41e3T + 9.12e5T^{2} \) |
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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
−8.758165875451702435532292292127, −8.043187340358688915225297526636, −6.74317372745553499999873563689, −6.13289043466406624192129568756, −5.22767461222078063757164759509, −4.54686553358942437178016167905, −3.52360879028872061928551112991, −2.45059689911978263152445739519, −1.69029688114168653335619848658, 0,
1.69029688114168653335619848658, 2.45059689911978263152445739519, 3.52360879028872061928551112991, 4.54686553358942437178016167905, 5.22767461222078063757164759509, 6.13289043466406624192129568756, 6.74317372745553499999873563689, 8.043187340358688915225297526636, 8.758165875451702435532292292127
