| L(s) = 1 | − 36.1·2-s − 228.·3-s + 796.·4-s + 2.34e3·5-s + 8.26e3·6-s + 528.·7-s − 1.02e4·8-s + 3.25e4·9-s − 8.49e4·10-s − 5.12e4·11-s − 1.82e5·12-s + 7.38e4·13-s − 1.91e4·14-s − 5.37e5·15-s − 3.57e4·16-s + 4.54e5·17-s − 1.17e6·18-s + 2.96e5·19-s + 1.87e6·20-s − 1.20e5·21-s + 1.85e6·22-s − 2.55e6·23-s + 2.35e6·24-s + 3.56e6·25-s − 2.67e6·26-s − 2.94e6·27-s + 4.21e5·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 1.62·3-s + 1.55·4-s + 1.68·5-s + 2.60·6-s + 0.0832·7-s − 0.887·8-s + 1.65·9-s − 2.68·10-s − 1.05·11-s − 2.53·12-s + 0.716·13-s − 0.133·14-s − 2.73·15-s − 0.136·16-s + 1.31·17-s − 2.64·18-s + 0.522·19-s + 2.61·20-s − 0.135·21-s + 1.68·22-s − 1.90·23-s + 1.44·24-s + 1.82·25-s − 1.14·26-s − 1.06·27-s + 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.6109147553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6109147553\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 + 36.1T + 512T^{2} \) |
| 3 | \( 1 + 228.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.34e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 528.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 7.38e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.54e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.96e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.55e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.91e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.32e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.57e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.16e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 4.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.17e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.52e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.34e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.83e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.71e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.94e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.12e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.40e7T + 7.60e17T^{2} \) |
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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
−13.82401648417069888016217541387, −12.42929618798848794942012912147, −10.98346617152445415108335011603, −10.27542309703807343691753744853, −9.503730386803657867920707608645, −7.71887651987812182902457939475, −6.18095243689604023284927578376, −5.44557686155874929679729994510, −1.90346113389485286250135964975, −0.74080174588626834765805469414,
0.74080174588626834765805469414, 1.90346113389485286250135964975, 5.44557686155874929679729994510, 6.18095243689604023284927578376, 7.71887651987812182902457939475, 9.503730386803657867920707608645, 10.27542309703807343691753744853, 10.98346617152445415108335011603, 12.42929618798848794942012912147, 13.82401648417069888016217541387
