| L(s) = 1 | + 1.63e4·2-s + 7.68e6·3-s + 2.68e8·4-s + 6.10e9·5-s + 1.25e11·6-s − 1.48e12·7-s + 4.39e12·8-s − 9.59e12·9-s + 1.00e14·10-s − 1.49e15·11-s + 2.06e15·12-s − 1.87e16·13-s − 2.42e16·14-s + 4.68e16·15-s + 7.20e16·16-s − 5.52e17·17-s − 1.57e17·18-s − 3.63e18·19-s + 1.63e18·20-s − 1.13e19·21-s − 2.44e19·22-s + 1.19e19·23-s + 3.37e19·24-s + 3.72e19·25-s − 3.07e20·26-s − 6.01e20·27-s − 3.97e20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.927·3-s + 0.5·4-s + 0.447·5-s + 0.655·6-s − 0.825·7-s + 0.353·8-s − 0.139·9-s + 0.316·10-s − 1.18·11-s + 0.463·12-s − 1.32·13-s − 0.583·14-s + 0.414·15-s + 0.250·16-s − 0.795·17-s − 0.0988·18-s − 1.04·19-s + 0.223·20-s − 0.765·21-s − 0.837·22-s + 0.214·23-s + 0.327·24-s + 0.200·25-s − 0.934·26-s − 1.05·27-s − 0.412·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.63e4T \) |
| 5 | \( 1 - 6.10e9T \) |
| good | 3 | \( 1 - 7.68e6T + 6.86e13T^{2} \) |
| 7 | \( 1 + 1.48e12T + 3.21e24T^{2} \) |
| 11 | \( 1 + 1.49e15T + 1.58e30T^{2} \) |
| 13 | \( 1 + 1.87e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + 5.52e17T + 4.81e35T^{2} \) |
| 19 | \( 1 + 3.63e18T + 1.21e37T^{2} \) |
| 23 | \( 1 - 1.19e19T + 3.09e39T^{2} \) |
| 29 | \( 1 - 1.49e21T + 2.56e42T^{2} \) |
| 31 | \( 1 - 2.12e20T + 1.77e43T^{2} \) |
| 37 | \( 1 - 4.36e22T + 3.00e45T^{2} \) |
| 41 | \( 1 - 1.24e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 4.89e23T + 2.34e47T^{2} \) |
| 47 | \( 1 - 3.20e24T + 3.09e48T^{2} \) |
| 53 | \( 1 + 1.61e25T + 1.00e50T^{2} \) |
| 59 | \( 1 + 6.74e25T + 2.26e51T^{2} \) |
| 61 | \( 1 + 4.59e25T + 5.95e51T^{2} \) |
| 67 | \( 1 + 3.04e26T + 9.04e52T^{2} \) |
| 71 | \( 1 - 8.77e25T + 4.85e53T^{2} \) |
| 73 | \( 1 + 1.54e27T + 1.08e54T^{2} \) |
| 79 | \( 1 - 8.64e25T + 1.07e55T^{2} \) |
| 83 | \( 1 - 8.92e27T + 4.50e55T^{2} \) |
| 89 | \( 1 - 4.85e27T + 3.40e56T^{2} \) |
| 97 | \( 1 - 6.50e28T + 4.13e57T^{2} \) |
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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
−13.47122809353914777495695037298, −12.54520928142389245067565693040, −10.57132827178239129074568349289, −9.204904545039867475866760619801, −7.64516201288252854182410150876, −6.12825567625343865066301396521, −4.59964177228313023993288801237, −2.88650740955351549416613284078, −2.32884995884816183750435443939, 0,
2.32884995884816183750435443939, 2.88650740955351549416613284078, 4.59964177228313023993288801237, 6.12825567625343865066301396521, 7.64516201288252854182410150876, 9.204904545039867475866760619801, 10.57132827178239129074568349289, 12.54520928142389245067565693040, 13.47122809353914777495695037298
