| L(s) = 1 | − 15.7·2-s − 22.0·3-s + 119.·4-s + 347.·6-s + 343·7-s + 137.·8-s − 1.69e3·9-s + 4.25e3·11-s − 2.63e3·12-s + 1.01e4·13-s − 5.39e3·14-s − 1.74e4·16-s + 7.93e3·17-s + 2.67e4·18-s + 2.38e4·19-s − 7.57e3·21-s − 6.68e4·22-s + 7.08e4·23-s − 3.04e3·24-s − 1.59e5·26-s + 8.58e4·27-s + 4.08e4·28-s − 2.41e5·29-s + 1.00e5·31-s + 2.56e5·32-s − 9.39e4·33-s − 1.24e5·34-s + ⋯ |
| L(s) = 1 | − 1.38·2-s − 0.472·3-s + 0.931·4-s + 0.656·6-s + 0.377·7-s + 0.0951·8-s − 0.776·9-s + 0.963·11-s − 0.439·12-s + 1.27·13-s − 0.525·14-s − 1.06·16-s + 0.391·17-s + 1.07·18-s + 0.797·19-s − 0.178·21-s − 1.33·22-s + 1.21·23-s − 0.0449·24-s − 1.77·26-s + 0.839·27-s + 0.352·28-s − 1.84·29-s + 0.608·31-s + 1.38·32-s − 0.454·33-s − 0.544·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8827661960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8827661960\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 15.7T + 128T^{2} \) |
| 3 | \( 1 + 22.0T + 2.18e3T^{2} \) |
| 11 | \( 1 - 4.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 7.93e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.38e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.08e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.85e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.54e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.21e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.99e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 9.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.34e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.37e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.80e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.09e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.21e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.02e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.25815870350686328330406736832, −10.41366312078375730682211778243, −9.149687365479496209061772260461, −8.648130442401984868694491110168, −7.49638984410160941352153789589, −6.39593476085470300861891048673, −5.16946819940153083339650610105, −3.49400745371452169277790367153, −1.62032864880688997028519857556, −0.70192485775403492972934186814,
0.70192485775403492972934186814, 1.62032864880688997028519857556, 3.49400745371452169277790367153, 5.16946819940153083339650610105, 6.39593476085470300861891048673, 7.49638984410160941352153789589, 8.648130442401984868694491110168, 9.149687365479496209061772260461, 10.41366312078375730682211778243, 11.25815870350686328330406736832
