| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s + 332.·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s + 2.65e3·10-s + 4.53e3·11-s − 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 8.97e3·15-s + 4.09e3·16-s − 1.68e3·17-s + 5.83e3·18-s + 4.30e4·19-s + 2.12e4·20-s + 9.26e3·21-s + 3.63e4·22-s + 8.00e4·23-s − 1.38e4·24-s + 3.22e4·25-s + 1.75e4·26-s − 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.18·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.840·10-s + 1.02·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.686·15-s + 0.250·16-s − 0.0831·17-s + 0.235·18-s + 1.43·19-s + 0.594·20-s + 0.218·21-s + 0.726·22-s + 1.37·23-s − 0.204·24-s + 0.413·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.607356910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.607356910\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 - 332.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 4.53e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.68e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.30e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.00e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.75e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.77e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.18e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.11e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.54e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.86e4T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.49e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 9.26e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.20e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.11e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.58e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.97e6T + 8.07e13T^{2} \) |
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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
−9.461696956133379235799850953533, −9.301092361062364837557995959743, −7.50199450256195138845662989740, −6.69320358521337146009370587670, −5.84109693147989222145431729794, −5.32666713063274193519957970774, −4.08994524974066139496114122415, −3.04241986630704641352208129291, −1.78512072902961486989476527460, −0.911088770615909371501490906501,
0.911088770615909371501490906501, 1.78512072902961486989476527460, 3.04241986630704641352208129291, 4.08994524974066139496114122415, 5.32666713063274193519957970774, 5.84109693147989222145431729794, 6.69320358521337146009370587670, 7.50199450256195138845662989740, 9.301092361062364837557995959743, 9.461696956133379235799850953533
