| L(s) = 1 | − 8·2-s − 82.6·3-s + 64·4-s − 228.·5-s + 661.·6-s − 449.·7-s − 512·8-s + 4.64e3·9-s + 1.82e3·10-s − 1.32e3·11-s − 5.29e3·12-s − 8.21e3·13-s + 3.59e3·14-s + 1.88e4·15-s + 4.09e3·16-s + 2.25e4·17-s − 3.71e4·18-s − 2.92e4·19-s − 1.46e4·20-s + 3.71e4·21-s + 1.05e4·22-s − 9.67e4·23-s + 4.23e4·24-s − 2.60e4·25-s + 6.57e4·26-s − 2.03e5·27-s − 2.87e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.816·5-s + 1.25·6-s − 0.495·7-s − 0.353·8-s + 2.12·9-s + 0.577·10-s − 0.299·11-s − 0.883·12-s − 1.03·13-s + 0.350·14-s + 1.44·15-s + 0.250·16-s + 1.11·17-s − 1.50·18-s − 0.978·19-s − 0.408·20-s + 0.876·21-s + 0.211·22-s − 1.65·23-s + 0.625·24-s − 0.333·25-s + 0.733·26-s − 1.98·27-s − 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 269 | \( 1 - 1.94e7T \) |
| good | 3 | \( 1 + 82.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 228.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 449.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.32e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.25e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.92e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.67e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.25e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 9.96e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.62e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.76e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.45e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.69e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.65e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.08e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.73e6T + 8.07e13T^{2} \) |
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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
−9.645844871967169834241509526037, −8.103246128127033904274412980882, −7.42788089894682414656340906366, −6.50315139656960914866137334508, −5.74623681147278097311658343939, −4.73768964142083637401290300871, −3.65824619068420940340202721663, −2.01910948530744845006006600608, −0.62010002493348291489827121907, 0,
0.62010002493348291489827121907, 2.01910948530744845006006600608, 3.65824619068420940340202721663, 4.73768964142083637401290300871, 5.74623681147278097311658343939, 6.50315139656960914866137334508, 7.42788089894682414656340906366, 8.103246128127033904274412980882, 9.645844871967169834241509526037
