| L(s) = 1 | − 199.·5-s + 343·7-s − 1.21e3·11-s + 6.44e3·13-s − 2.67e4·17-s + 1.49e4·19-s + 7.35e4·23-s − 3.84e4·25-s + 1.06e5·29-s + 6.68e4·31-s − 6.83e4·35-s − 4.79e5·37-s + 6.44e5·41-s + 1.45e5·43-s − 1.01e6·47-s + 1.17e5·49-s − 3.42e4·53-s + 2.42e5·55-s + 4.43e5·59-s − 1.14e6·61-s − 1.28e6·65-s − 4.31e6·67-s − 2.54e6·71-s − 3.67e6·73-s − 4.18e5·77-s + 8.55e6·79-s + 1.79e6·83-s + ⋯ |
| L(s) = 1 | − 0.712·5-s + 0.377·7-s − 0.276·11-s + 0.814·13-s − 1.32·17-s + 0.498·19-s + 1.25·23-s − 0.492·25-s + 0.810·29-s + 0.402·31-s − 0.269·35-s − 1.55·37-s + 1.46·41-s + 0.278·43-s − 1.42·47-s + 0.142·49-s − 0.0315·53-s + 0.196·55-s + 0.281·59-s − 0.643·61-s − 0.580·65-s − 1.75·67-s − 0.845·71-s − 1.10·73-s − 0.104·77-s + 1.95·79-s + 0.345·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 5 | \( 1 + 199.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.67e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.49e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.06e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.44e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.42e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.54e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.79e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.72e7T + 8.07e13T^{2} \) |
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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
−10.56465237177144979119831109850, −9.171669599220728907859902151180, −8.379402057985717868188270064799, −7.41502371030172150145270462484, −6.37919982814729788929000249761, −5.04540389190013401059660538677, −4.06205422636702563107491368285, −2.84191344202672135354304369289, −1.34345509340296335783139924662, 0,
1.34345509340296335783139924662, 2.84191344202672135354304369289, 4.06205422636702563107491368285, 5.04540389190013401059660538677, 6.37919982814729788929000249761, 7.41502371030172150145270462484, 8.379402057985717868188270064799, 9.171669599220728907859902151180, 10.56465237177144979119831109850
