| L(s) = 1 | − 15.5·2-s + 113.·4-s + 141.·5-s + 105.·7-s + 232.·8-s − 2.20e3·10-s + 555.·11-s − 1.12e4·13-s − 1.63e3·14-s − 1.80e4·16-s + 3.13e3·17-s − 2.19e4·19-s + 1.60e4·20-s − 8.61e3·22-s − 2.85e4·23-s − 5.80e4·25-s + 1.75e5·26-s + 1.19e4·28-s − 1.60e5·29-s − 1.63e5·31-s + 2.50e5·32-s − 4.87e4·34-s + 1.49e4·35-s − 1.31e5·37-s + 3.40e5·38-s + 3.29e4·40-s + 2.77e5·41-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.883·4-s + 0.507·5-s + 0.116·7-s + 0.160·8-s − 0.696·10-s + 0.125·11-s − 1.42·13-s − 0.159·14-s − 1.10·16-s + 0.155·17-s − 0.734·19-s + 0.448·20-s − 0.172·22-s − 0.489·23-s − 0.742·25-s + 1.95·26-s + 0.102·28-s − 1.22·29-s − 0.986·31-s + 1.35·32-s − 0.212·34-s + 0.0588·35-s − 0.425·37-s + 1.00·38-s + 0.0813·40-s + 0.627·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5696253529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5696253529\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 15.5T + 128T^{2} \) |
| 5 | \( 1 - 141.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 105.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 555.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.12e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.13e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.85e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.60e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.63e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.39e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.49e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 2.41e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.34e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.27e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.04e6T + 8.07e13T^{2} \) |
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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
−9.430311192387847207122249393214, −9.180129549731568877724332482692, −7.86691718492362526756528563042, −7.43420240370458302971297097653, −6.29145064108581303146379289376, −5.18643501352540033073809496268, −4.03438673817452235742614820686, −2.37582853765647136029115556677, −1.72457213820629353152334023600, −0.39648726521804082899851102717,
0.39648726521804082899851102717, 1.72457213820629353152334023600, 2.37582853765647136029115556677, 4.03438673817452235742614820686, 5.18643501352540033073809496268, 6.29145064108581303146379289376, 7.43420240370458302971297097653, 7.86691718492362526756528563042, 9.180129549731568877724332482692, 9.430311192387847207122249393214
