| L(s) = 1 | + 56.0·2-s + 4.54e3·3-s − 2.96e4·4-s − 1.74e5·5-s + 2.54e5·6-s + 2.13e6·7-s − 3.49e6·8-s + 6.26e6·9-s − 9.78e6·10-s − 3.19e7·11-s − 1.34e8·12-s + 4.37e7·13-s + 1.19e8·14-s − 7.93e8·15-s + 7.75e8·16-s + 2.04e9·17-s + 3.50e8·18-s + 1.93e9·19-s + 5.17e9·20-s + 9.67e9·21-s − 1.79e9·22-s + 2.26e10·23-s − 1.58e10·24-s + 5.81e6·25-s + 2.45e9·26-s − 3.67e10·27-s − 6.31e10·28-s + ⋯ |
| L(s) = 1 | + 0.309·2-s + 1.19·3-s − 0.904·4-s − 1.00·5-s + 0.370·6-s + 0.977·7-s − 0.589·8-s + 0.436·9-s − 0.309·10-s − 0.494·11-s − 1.08·12-s + 0.193·13-s + 0.302·14-s − 1.19·15-s + 0.721·16-s + 1.21·17-s + 0.135·18-s + 0.496·19-s + 0.904·20-s + 1.17·21-s − 0.153·22-s + 1.38·23-s − 0.706·24-s + 0.000190·25-s + 0.0598·26-s − 0.675·27-s − 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.660296080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.660296080\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 1.72e10T \) |
| good | 2 | \( 1 - 56.0T + 3.27e4T^{2} \) |
| 3 | \( 1 - 4.54e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.74e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.13e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 3.19e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 4.37e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.04e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.93e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.26e10T + 2.66e20T^{2} \) |
| 31 | \( 1 - 1.91e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 3.69e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 9.13e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.38e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.53e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.41e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.58e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 1.33e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 9.69e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 5.31e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.16e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 5.19e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 2.76e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 2.68e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 5.44e14T + 6.33e29T^{2} \) |
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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
−13.95849136576899945207940281138, −12.70062347196969183841268173442, −11.34016954002898671374550460933, −9.552272835019214311687836471821, −8.285647237964252725705046561508, −7.77155055947035796533685677615, −5.21506192896945736102939723914, −3.97236936754188472682188769702, −2.90335853861310401112136038931, −0.904466863186773059573061754940,
0.904466863186773059573061754940, 2.90335853861310401112136038931, 3.97236936754188472682188769702, 5.21506192896945736102939723914, 7.77155055947035796533685677615, 8.285647237964252725705046561508, 9.552272835019214311687836471821, 11.34016954002898671374550460933, 12.70062347196969183841268173442, 13.95849136576899945207940281138
