| L(s) = 1 | − 1.35e18·2-s + 1.15e28·3-s − 8.81e36·4-s + 1.19e42·5-s − 1.55e46·6-s + 4.29e51·7-s + 2.62e55·8-s − 4.83e58·9-s − 1.60e60·10-s − 6.95e63·11-s − 1.01e65·12-s + 4.61e68·13-s − 5.79e69·14-s + 1.37e70·15-s + 5.82e73·16-s − 3.52e75·17-s + 6.53e76·18-s + 5.44e77·19-s − 1.04e79·20-s + 4.95e79·21-s + 9.38e81·22-s − 5.17e83·23-s + 3.03e83·24-s − 9.26e85·25-s − 6.23e86·26-s − 1.11e87·27-s − 3.78e88·28-s + ⋯ |
| L(s) = 1 | − 0.413·2-s + 0.0524·3-s − 0.828·4-s + 0.122·5-s − 0.0217·6-s + 0.455·7-s + 0.757·8-s − 0.997·9-s − 0.0508·10-s − 0.626·11-s − 0.0434·12-s + 1.43·13-s − 0.188·14-s + 0.00643·15-s + 0.515·16-s − 0.749·17-s + 0.412·18-s + 0.123·19-s − 0.101·20-s + 0.0239·21-s + 0.259·22-s − 0.927·23-s + 0.0397·24-s − 0.984·25-s − 0.593·26-s − 0.104·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(124-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(62)\) |
\(\approx\) |
\(1.010741257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.010741257\) |
| \(L(\frac{125}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.35e18T + 1.06e37T^{2} \) |
| 3 | \( 1 - 1.15e28T + 4.85e58T^{2} \) |
| 5 | \( 1 - 1.19e42T + 9.40e85T^{2} \) |
| 7 | \( 1 - 4.29e51T + 8.85e103T^{2} \) |
| 11 | \( 1 + 6.95e63T + 1.23e128T^{2} \) |
| 13 | \( 1 - 4.61e68T + 1.03e137T^{2} \) |
| 17 | \( 1 + 3.52e75T + 2.21e151T^{2} \) |
| 19 | \( 1 - 5.44e77T + 1.93e157T^{2} \) |
| 23 | \( 1 + 5.17e83T + 3.10e167T^{2} \) |
| 29 | \( 1 - 8.59e89T + 7.49e179T^{2} \) |
| 31 | \( 1 - 2.75e91T + 2.73e183T^{2} \) |
| 37 | \( 1 + 2.58e96T + 7.74e192T^{2} \) |
| 41 | \( 1 - 4.18e98T + 2.35e198T^{2} \) |
| 43 | \( 1 + 2.45e100T + 8.25e200T^{2} \) |
| 47 | \( 1 + 6.57e102T + 4.65e205T^{2} \) |
| 53 | \( 1 + 2.38e105T + 1.21e212T^{2} \) |
| 59 | \( 1 - 8.41e108T + 6.52e217T^{2} \) |
| 61 | \( 1 - 2.09e109T + 3.94e219T^{2} \) |
| 67 | \( 1 + 1.87e112T + 4.04e224T^{2} \) |
| 71 | \( 1 - 1.00e114T + 5.06e227T^{2} \) |
| 73 | \( 1 + 2.35e114T + 1.54e229T^{2} \) |
| 79 | \( 1 - 1.72e115T + 2.55e233T^{2} \) |
| 83 | \( 1 + 4.88e117T + 1.11e236T^{2} \) |
| 89 | \( 1 - 6.06e119T + 5.95e239T^{2} \) |
| 97 | \( 1 - 2.90e122T + 2.36e244T^{2} \) |
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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
−13.31478141127581572295887645045, −11.47597392838985944881322292079, −10.20001070563626795841254394962, −8.712397409043619577304098012692, −8.068040539822525286398678559854, −6.08163254416893621577162849035, −4.84201460262990549435878727665, −3.49607860895428953728423275397, −1.87776249789926123512925351803, −0.52393524992424443509180332017,
0.52393524992424443509180332017, 1.87776249789926123512925351803, 3.49607860895428953728423275397, 4.84201460262990549435878727665, 6.08163254416893621577162849035, 8.068040539822525286398678559854, 8.712397409043619577304098012692, 10.20001070563626795841254394962, 11.47597392838985944881322292079, 13.31478141127581572295887645045
