| L(s) = 1 | + 7.72e16·2-s + 5.46e26·3-s + 3.37e33·4-s + 2.64e38·5-s + 4.22e43·6-s + 1.15e47·7-s + 6.03e49·8-s + 2.07e53·9-s + 2.04e55·10-s + 1.44e57·11-s + 1.84e60·12-s + 7.18e61·13-s + 8.91e63·14-s + 1.44e65·15-s − 4.10e66·16-s − 2.34e68·17-s + 1.59e70·18-s − 1.51e71·19-s + 8.91e71·20-s + 6.30e73·21-s + 1.11e74·22-s − 3.38e75·23-s + 3.29e76·24-s − 3.15e77·25-s + 5.55e78·26-s + 6.31e79·27-s + 3.89e80·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.80·3-s + 1.30·4-s + 0.425·5-s + 2.74·6-s + 1.44·7-s + 0.456·8-s + 2.26·9-s + 0.645·10-s + 0.229·11-s + 2.35·12-s + 1.07·13-s + 2.18·14-s + 0.768·15-s − 0.608·16-s − 1.20·17-s + 3.43·18-s − 1.61·19-s + 0.553·20-s + 2.60·21-s + 0.348·22-s − 0.898·23-s + 0.824·24-s − 0.819·25-s + 1.63·26-s + 2.29·27-s + 1.87·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(112-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+55.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(56)\) |
\(\approx\) |
\(12.21193708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.21193708\) |
| \(L(\frac{113}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 7.72e16T + 2.59e33T^{2} \) |
| 3 | \( 1 - 5.46e26T + 9.12e52T^{2} \) |
| 5 | \( 1 - 2.64e38T + 3.85e77T^{2} \) |
| 7 | \( 1 - 1.15e47T + 6.39e93T^{2} \) |
| 11 | \( 1 - 1.44e57T + 3.93e115T^{2} \) |
| 13 | \( 1 - 7.18e61T + 4.44e123T^{2} \) |
| 17 | \( 1 + 2.34e68T + 3.80e136T^{2} \) |
| 19 | \( 1 + 1.51e71T + 8.74e141T^{2} \) |
| 23 | \( 1 + 3.38e75T + 1.41e151T^{2} \) |
| 29 | \( 1 + 3.19e80T + 2.11e162T^{2} \) |
| 31 | \( 1 + 3.12e82T + 3.47e165T^{2} \) |
| 37 | \( 1 - 1.41e87T + 1.17e174T^{2} \) |
| 41 | \( 1 - 2.68e89T + 1.04e179T^{2} \) |
| 43 | \( 1 - 1.11e90T + 2.06e181T^{2} \) |
| 47 | \( 1 + 2.44e92T + 4.00e185T^{2} \) |
| 53 | \( 1 + 2.55e95T + 2.48e191T^{2} \) |
| 59 | \( 1 - 7.60e96T + 3.66e196T^{2} \) |
| 61 | \( 1 + 1.41e99T + 1.48e198T^{2} \) |
| 67 | \( 1 - 7.85e100T + 4.94e202T^{2} \) |
| 71 | \( 1 - 3.19e102T + 3.08e205T^{2} \) |
| 73 | \( 1 - 2.79e103T + 6.74e206T^{2} \) |
| 79 | \( 1 - 8.38e104T + 4.33e210T^{2} \) |
| 83 | \( 1 + 2.05e106T + 1.04e213T^{2} \) |
| 89 | \( 1 - 1.94e108T + 2.41e216T^{2} \) |
| 97 | \( 1 + 1.03e110T + 3.40e220T^{2} \) |
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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
−13.83665141226029056248166859665, −12.94889310246105981768438399445, −11.08221755201092160128433163783, −9.018032313652523799171197301625, −7.971898218030920337434823920803, −6.25725088549338865599615230866, −4.47844902041698024924091671533, −3.84776362764604722844209540291, −2.32474425458589040657473981174, −1.78662405374143689261168904758,
1.78662405374143689261168904758, 2.32474425458589040657473981174, 3.84776362764604722844209540291, 4.47844902041698024924091671533, 6.25725088549338865599615230866, 7.971898218030920337434823920803, 9.018032313652523799171197301625, 11.08221755201092160128433163783, 12.94889310246105981768438399445, 13.83665141226029056248166859665
