| L(s) = 1 | − 8.08·2-s − 62.6·4-s − 164.·5-s + 1.31e3·7-s + 1.54e3·8-s + 1.33e3·10-s − 8.63e3·11-s + 8.42e3·13-s − 1.06e4·14-s − 4.43e3·16-s − 6.30e3·17-s + 1.66e4·19-s + 1.03e4·20-s + 6.97e4·22-s − 4.41e4·23-s − 5.09e4·25-s − 6.80e4·26-s − 8.24e4·28-s + 2.38e4·29-s + 2.71e4·31-s − 1.61e5·32-s + 5.09e4·34-s − 2.16e5·35-s − 1.14e5·37-s − 1.34e5·38-s − 2.53e5·40-s + 7.92e5·41-s + ⋯ |
| L(s) = 1 | − 0.714·2-s − 0.489·4-s − 0.589·5-s + 1.44·7-s + 1.06·8-s + 0.420·10-s − 1.95·11-s + 1.06·13-s − 1.03·14-s − 0.270·16-s − 0.311·17-s + 0.557·19-s + 0.288·20-s + 1.39·22-s − 0.755·23-s − 0.652·25-s − 0.759·26-s − 0.709·28-s + 0.181·29-s + 0.163·31-s − 0.870·32-s + 0.222·34-s − 0.853·35-s − 0.370·37-s − 0.398·38-s − 0.627·40-s + 1.79·41-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 | \( 1 + 8.08T + 128T^{2} \) |
| 5 | \( 1 + 164.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.31e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 8.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.42e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 6.30e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.66e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.41e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.38e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.71e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.14e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.92e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 4.90e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.64e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 6.25e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.26e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 9.34e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.08e6T + 8.07e13T^{2} \) |
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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
−9.666150994971542860056275773008, −8.398851047190612297179713207061, −8.094244610347101345656545618528, −7.42755062721062665994641271108, −5.62811456650655198962951628518, −4.81668535499910713495278606753, −3.87425450837297080052677349370, −2.25802600719392982546665802400, −1.06811791037677096247122582780, 0,
1.06811791037677096247122582780, 2.25802600719392982546665802400, 3.87425450837297080052677349370, 4.81668535499910713495278606753, 5.62811456650655198962951628518, 7.42755062721062665994641271108, 8.094244610347101345656545618528, 8.398851047190612297179713207061, 9.666150994971542860056275773008
