| L(s) = 1 | + 7.02·2-s − 27·3-s − 78.6·4-s − 266.·5-s − 189.·6-s + 665.·7-s − 1.45e3·8-s + 729·9-s − 1.87e3·10-s + 2.99e3·11-s + 2.12e3·12-s + 1.11e4·13-s + 4.67e3·14-s + 7.20e3·15-s − 120.·16-s + 1.73e4·17-s + 5.11e3·18-s − 7.56e3·19-s + 2.09e4·20-s − 1.79e4·21-s + 2.10e4·22-s − 8.48e4·23-s + 3.91e4·24-s − 6.95e3·25-s + 7.80e4·26-s − 1.96e4·27-s − 5.23e4·28-s + ⋯ |
| L(s) = 1 | + 0.620·2-s − 0.577·3-s − 0.614·4-s − 0.954·5-s − 0.358·6-s + 0.733·7-s − 1.00·8-s + 0.333·9-s − 0.592·10-s + 0.678·11-s + 0.354·12-s + 1.40·13-s + 0.455·14-s + 0.551·15-s − 0.00733·16-s + 0.855·17-s + 0.206·18-s − 0.253·19-s + 0.586·20-s − 0.423·21-s + 0.421·22-s − 1.45·23-s + 0.578·24-s − 0.0890·25-s + 0.870·26-s − 0.192·27-s − 0.450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 7.02T + 128T^{2} \) |
| 5 | \( 1 + 266.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 665.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.11e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.56e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 5.97e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.62e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.75e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.08e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.51e4T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.31e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.04e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.53e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.47e7T + 8.07e13T^{2} \) |
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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
−11.29457124482352558825937177982, −10.01693090855648701576993789383, −8.655698597064306907605014713134, −7.922315510076863896815162714471, −6.39592128582760935861291458078, −5.39749936419377176210052277549, −4.20782567262960017636619195105, −3.60362322088298195603989508758, −1.32950575629041538734998936575, 0,
1.32950575629041538734998936575, 3.60362322088298195603989508758, 4.20782567262960017636619195105, 5.39749936419377176210052277549, 6.39592128582760935861291458078, 7.922315510076863896815162714471, 8.655698597064306907605014713134, 10.01693090855648701576993789383, 11.29457124482352558825937177982
