| L(s) = 1 | − 7.80·2-s − 2.71·3-s − 67.0·4-s − 402.·5-s + 21.1·6-s − 356.·7-s + 1.52e3·8-s − 2.17e3·9-s + 3.13e3·10-s + 338.·11-s + 181.·12-s − 1.90e3·13-s + 2.78e3·14-s + 1.09e3·15-s − 3.30e3·16-s − 1.37e4·17-s + 1.70e4·18-s + 3.23e4·19-s + 2.69e4·20-s + 968.·21-s − 2.64e3·22-s + 1.03e5·23-s − 4.13e3·24-s + 8.34e4·25-s + 1.48e4·26-s + 1.18e4·27-s + 2.39e4·28-s + ⋯ |
| L(s) = 1 | − 0.689·2-s − 0.0580·3-s − 0.523·4-s − 1.43·5-s + 0.0400·6-s − 0.393·7-s + 1.05·8-s − 0.996·9-s + 0.992·10-s + 0.0767·11-s + 0.0304·12-s − 0.240·13-s + 0.271·14-s + 0.0834·15-s − 0.201·16-s − 0.676·17-s + 0.687·18-s + 1.08·19-s + 0.753·20-s + 0.0228·21-s − 0.0529·22-s + 1.76·23-s − 0.0610·24-s + 1.06·25-s + 0.165·26-s + 0.115·27-s + 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.4526656375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4526656375\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + 7.95e4T \) |
| good | 2 | \( 1 + 7.80T + 128T^{2} \) |
| 3 | \( 1 + 2.71T + 2.18e3T^{2} \) |
| 5 | \( 1 + 402.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 356.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 338.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.90e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.37e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.03e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 97.3T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.91e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.80e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.80e4T + 1.94e11T^{2} \) |
| 47 | \( 1 - 1.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.50e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.16e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.63e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.56e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.01e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.90e6T + 8.07e13T^{2} \) |
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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
−14.56571582676339453962771164518, −13.23397514502078934662607528022, −11.84994635825642472381815066397, −10.86106503956554879124408322513, −9.268500937091303074358541706051, −8.316925573805999713561220937137, −7.15307244827558591904449538085, −4.95111035611260849295831710547, −3.38232308648273880461724115570, −0.55235583804325382276868960182,
0.55235583804325382276868960182, 3.38232308648273880461724115570, 4.95111035611260849295831710547, 7.15307244827558591904449538085, 8.316925573805999713561220937137, 9.268500937091303074358541706051, 10.86106503956554879124408322513, 11.84994635825642472381815066397, 13.23397514502078934662607528022, 14.56571582676339453962771164518
