L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 506.·5-s − 216·6-s − 836.·7-s − 512·8-s + 729·9-s + 4.05e3·10-s + 3.37e3·11-s + 1.72e3·12-s − 3.44e3·13-s + 6.68e3·14-s − 1.36e4·15-s + 4.09e3·16-s − 3.24e3·17-s − 5.83e3·18-s − 7.60e3·19-s − 3.24e4·20-s − 2.25e4·21-s − 2.70e4·22-s + 1.04e5·23-s − 1.38e4·24-s + 1.78e5·25-s + 2.75e4·26-s + 1.96e4·27-s − 5.35e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.81·5-s − 0.408·6-s − 0.921·7-s − 0.353·8-s + 0.333·9-s + 1.28·10-s + 0.765·11-s + 0.288·12-s − 0.434·13-s + 0.651·14-s − 1.04·15-s + 0.250·16-s − 0.160·17-s − 0.235·18-s − 0.254·19-s − 0.906·20-s − 0.531·21-s − 0.541·22-s + 1.78·23-s − 0.204·24-s + 2.28·25-s + 0.307·26-s + 0.192·27-s − 0.460·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 + 2.05e5T \) |
good | 5 | \( 1 + 506.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 836.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.44e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.24e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 7.60e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.04e5T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.30e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.34e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.64e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.50e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.84e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.85e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.01e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.34e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.43e6T + 8.07e13T^{2} \) |
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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
−9.416064321043271438328427344901, −9.023620140426397519009928755257, −7.84264111521574374058129732707, −7.32206795589136678314599193906, −6.38567225716750141775386817517, −4.53179259142080000717820134205, −3.58027609163301513316299285228, −2.75496191922328816353634627861, −1.00506809067704478133099818718, 0,
1.00506809067704478133099818718, 2.75496191922328816353634627861, 3.58027609163301513316299285228, 4.53179259142080000717820134205, 6.38567225716750141775386817517, 7.32206795589136678314599193906, 7.84264111521574374058129732707, 9.023620140426397519009928755257, 9.416064321043271438328427344901