| L(s) = 1 | − 12.5·2-s + 28.4·4-s − 71.4·5-s + 1.02e3·7-s + 1.24e3·8-s + 893.·10-s + 5.74e3·11-s + 6.11e3·13-s − 1.27e4·14-s − 1.92e4·16-s − 2.12e4·17-s − 2.27e4·19-s − 2.03e3·20-s − 7.18e4·22-s + 4.68e4·23-s − 7.30e4·25-s − 7.65e4·26-s + 2.90e4·28-s − 2.21e5·29-s − 3.43e3·31-s + 8.09e4·32-s + 2.65e5·34-s − 7.28e4·35-s − 2.38e5·37-s + 2.84e5·38-s − 8.89e4·40-s + 5.53e5·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.222·4-s − 0.255·5-s + 1.12·7-s + 0.859·8-s + 0.282·10-s + 1.30·11-s + 0.772·13-s − 1.24·14-s − 1.17·16-s − 1.04·17-s − 0.760·19-s − 0.0567·20-s − 1.43·22-s + 0.803·23-s − 0.934·25-s − 0.853·26-s + 0.249·28-s − 1.68·29-s − 0.0207·31-s + 0.436·32-s + 1.15·34-s − 0.287·35-s − 0.775·37-s + 0.840·38-s − 0.219·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 12.5T + 128T^{2} \) |
| 5 | \( 1 + 71.4T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.02e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.74e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.11e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.27e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.68e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.21e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.43e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.38e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.53e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.67e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.17e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.28e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.64e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.25e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.60e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.03e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.12e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.33e6T + 8.07e13T^{2} \) |
| 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
−9.213033247489703771135015918791, −9.001977592797067027676737963904, −7.993016404882878947131928663316, −7.22103327267403443927803159713, −6.05718315783038086768834473650, −4.60325336642073083119002744283, −3.90233030517937669502403006570, −1.95199799975233247867414827668, −1.24110465153047932436082249546, 0,
1.24110465153047932436082249546, 1.95199799975233247867414827668, 3.90233030517937669502403006570, 4.60325336642073083119002744283, 6.05718315783038086768834473650, 7.22103327267403443927803159713, 7.993016404882878947131928663316, 9.001977592797067027676737963904, 9.213033247489703771135015918791
