| L(s) = 1 | − 62.7·2-s + 761.·3-s + 1.89e3·4-s + 1.40e3·5-s − 4.78e4·6-s + 7.74e4·7-s + 9.90e3·8-s + 4.03e5·9-s − 8.78e4·10-s + 4.02e5·11-s + 1.43e6·12-s − 1.71e6·13-s − 4.85e6·14-s + 1.06e6·15-s − 4.49e6·16-s + 3.45e5·17-s − 2.52e7·18-s + 1.40e7·19-s + 2.64e6·20-s + 5.89e7·21-s − 2.52e7·22-s − 5.18e7·23-s + 7.54e6·24-s − 4.68e7·25-s + 1.07e8·26-s + 1.72e8·27-s + 1.46e8·28-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 1.80·3-s + 0.922·4-s + 0.200·5-s − 2.50·6-s + 1.74·7-s + 0.106·8-s + 2.27·9-s − 0.277·10-s + 0.753·11-s + 1.67·12-s − 1.28·13-s − 2.41·14-s + 0.362·15-s − 1.07·16-s + 0.0590·17-s − 3.15·18-s + 1.30·19-s + 0.184·20-s + 3.15·21-s − 1.04·22-s − 1.67·23-s + 0.193·24-s − 0.959·25-s + 1.78·26-s + 2.30·27-s + 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.300823657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300823657\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 62.7T + 2.04e3T^{2} \) |
| 3 | \( 1 - 761.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 1.40e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 7.74e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 4.02e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.71e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.45e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.40e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.18e7T + 9.52e14T^{2} \) |
| 31 | \( 1 - 1.93e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.45e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 4.34e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 6.86e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.56e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.41e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.19e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 3.42e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 3.42e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 8.70e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.70e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.47e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.31e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.02e11T + 7.15e21T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.38960866969819922941026291104, −13.95212913322460658649078027834, −11.78308446778503209997058146741, −10.02358115945096245641900493898, −9.226398720030916792788166990719, −8.002501916800134425796904764219, −7.58393674998655304536762807662, −4.39169208106950592999066683440, −2.26881026006887646156827688382, −1.36773128678383788277628436436,
1.36773128678383788277628436436, 2.26881026006887646156827688382, 4.39169208106950592999066683440, 7.58393674998655304536762807662, 8.002501916800134425796904764219, 9.226398720030916792788166990719, 10.02358115945096245641900493898, 11.78308446778503209997058146741, 13.95212913322460658649078027834, 14.38960866969819922941026291104
