| L(s) = 1 | − 1.95e18·2-s − 1.11e29·3-s + 1.14e36·4-s + 9.20e41·5-s + 2.18e47·6-s + 2.38e51·7-s + 2.94e54·8-s + 7.10e57·9-s − 1.79e60·10-s − 1.41e63·11-s − 1.28e65·12-s − 1.39e67·13-s − 4.65e69·14-s − 1.02e71·15-s − 8.80e72·16-s − 1.04e74·17-s − 1.38e76·18-s + 3.51e77·19-s + 1.05e78·20-s − 2.66e80·21-s + 2.75e81·22-s + 6.21e81·23-s − 3.29e83·24-s − 2.91e84·25-s + 2.72e85·26-s − 1.91e86·27-s + 2.73e87·28-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 1.52·3-s + 0.432·4-s + 0.474·5-s + 1.82·6-s + 1.77·7-s + 0.679·8-s + 1.31·9-s − 0.567·10-s − 1.39·11-s − 0.657·12-s − 0.564·13-s − 2.12·14-s − 0.722·15-s − 1.24·16-s − 0.377·17-s − 1.57·18-s + 1.51·19-s + 0.205·20-s − 2.70·21-s + 1.67·22-s + 0.256·23-s − 1.03·24-s − 0.774·25-s + 0.675·26-s − 0.482·27-s + 0.766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(122-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+60.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(61)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{123}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.95e18T + 2.65e36T^{2} \) |
| 3 | \( 1 + 1.11e29T + 5.39e57T^{2} \) |
| 5 | \( 1 - 9.20e41T + 3.76e84T^{2} \) |
| 7 | \( 1 - 2.38e51T + 1.80e102T^{2} \) |
| 11 | \( 1 + 1.41e63T + 1.01e126T^{2} \) |
| 13 | \( 1 + 1.39e67T + 6.12e134T^{2} \) |
| 17 | \( 1 + 1.04e74T + 7.66e148T^{2} \) |
| 19 | \( 1 - 3.51e77T + 5.36e154T^{2} \) |
| 23 | \( 1 - 6.21e81T + 5.87e164T^{2} \) |
| 29 | \( 1 - 8.72e87T + 8.91e176T^{2} \) |
| 31 | \( 1 - 8.16e89T + 2.84e180T^{2} \) |
| 37 | \( 1 + 1.06e95T + 5.65e189T^{2} \) |
| 41 | \( 1 + 6.03e97T + 1.40e195T^{2} \) |
| 43 | \( 1 - 7.50e97T + 4.46e197T^{2} \) |
| 47 | \( 1 - 1.34e100T + 2.10e202T^{2} \) |
| 53 | \( 1 + 1.68e103T + 4.33e208T^{2} \) |
| 59 | \( 1 + 1.64e107T + 1.87e214T^{2} \) |
| 61 | \( 1 - 1.82e108T + 1.05e216T^{2} \) |
| 67 | \( 1 - 4.98e110T + 9.01e220T^{2} \) |
| 71 | \( 1 - 3.59e110T + 1.00e224T^{2} \) |
| 73 | \( 1 + 6.05e112T + 2.89e225T^{2} \) |
| 79 | \( 1 - 4.29e114T + 4.10e229T^{2} \) |
| 83 | \( 1 - 1.05e116T + 1.61e232T^{2} \) |
| 89 | \( 1 - 2.40e117T + 7.51e235T^{2} \) |
| 97 | \( 1 + 1.86e120T + 2.50e240T^{2} \) |
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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
−11.82829530351467842053008196721, −10.85889135779727275157053920192, −9.986958828496370273547646139711, −8.244951050510306835749225451587, −7.21220343889138741949800387871, −5.30924204889092104081337876836, −4.88531097672066600691003355400, −2.02206111376412570107210411601, −1.03367236365577053618719383018, 0,
1.03367236365577053618719383018, 2.02206111376412570107210411601, 4.88531097672066600691003355400, 5.30924204889092104081337876836, 7.21220343889138741949800387871, 8.244951050510306835749225451587, 9.986958828496370273547646139711, 10.85889135779727275157053920192, 11.82829530351467842053008196721
