| L(s) = 1 | − 8·2-s + 55.2·3-s + 64·4-s − 82.2·5-s − 441.·6-s + 195.·7-s − 512·8-s + 863.·9-s + 658.·10-s − 3.90e3·11-s + 3.53e3·12-s − 1.48e4·13-s − 1.56e3·14-s − 4.54e3·15-s + 4.09e3·16-s − 6.73e3·17-s − 6.90e3·18-s + 3.69e4·19-s − 5.26e3·20-s + 1.07e4·21-s + 3.12e4·22-s + 4.62e4·23-s − 2.82e4·24-s − 7.13e4·25-s + 1.18e5·26-s − 7.31e4·27-s + 1.25e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.18·3-s + 0.5·4-s − 0.294·5-s − 0.835·6-s + 0.215·7-s − 0.353·8-s + 0.394·9-s + 0.208·10-s − 0.883·11-s + 0.590·12-s − 1.87·13-s − 0.152·14-s − 0.347·15-s + 0.250·16-s − 0.332·17-s − 0.279·18-s + 1.23·19-s − 0.147·20-s + 0.254·21-s + 0.624·22-s + 0.793·23-s − 0.417·24-s − 0.913·25-s + 1.32·26-s − 0.714·27-s + 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 37 | \( 1 - 5.06e4T \) |
| good | 3 | \( 1 - 55.2T + 2.18e3T^{2} \) |
| 5 | \( 1 + 82.2T + 7.81e4T^{2} \) |
| 7 | \( 1 - 195.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.48e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.73e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.69e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.62e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.00e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.05e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 6.70e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.20e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.15e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.88e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.06e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.27e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.23e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.65e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.97e5T + 8.07e13T^{2} \) |
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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
−12.57131314614407240130830775474, −11.34944216406116872741069675876, −9.962508126897688723390297842405, −9.130898114351751010074685737206, −7.893191366413452225076333575503, −7.31232586276366107171470739243, −5.13599974824715060865115092151, −3.18869547090167069367072090705, −2.09273615838921581829043172705, 0,
2.09273615838921581829043172705, 3.18869547090167069367072090705, 5.13599974824715060865115092151, 7.31232586276366107171470739243, 7.893191366413452225076333575503, 9.130898114351751010074685737206, 9.962508126897688723390297842405, 11.34944216406116872741069675876, 12.57131314614407240130830775474
