| L(s) = 1 | − 3.15e18·2-s + 6.48e28·3-s + 7.29e36·4-s − 2.23e41·5-s − 2.04e47·6-s + 1.98e51·7-s − 1.46e55·8-s − 1.18e57·9-s + 7.06e59·10-s + 1.11e63·11-s + 4.73e65·12-s + 2.72e67·13-s − 6.26e69·14-s − 1.45e70·15-s + 2.67e73·16-s − 1.03e74·17-s + 3.74e75·18-s − 2.25e77·19-s − 1.63e78·20-s + 1.28e80·21-s − 3.52e81·22-s − 1.89e82·23-s − 9.48e83·24-s − 3.71e84·25-s − 8.59e85·26-s − 4.26e86·27-s + 1.44e88·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.883·3-s + 2.74·4-s − 0.115·5-s − 1.70·6-s + 1.47·7-s − 3.37·8-s − 0.220·9-s + 0.223·10-s + 1.10·11-s + 2.42·12-s + 1.10·13-s − 2.86·14-s − 0.101·15-s + 3.78·16-s − 0.372·17-s + 0.425·18-s − 0.973·19-s − 0.316·20-s + 1.30·21-s − 2.13·22-s − 0.782·23-s − 2.98·24-s − 0.986·25-s − 2.13·26-s − 1.07·27-s + 4.05·28-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 | \( 1 + 3.15e18T + 2.65e36T^{2} \) |
| 3 | \( 1 - 6.48e28T + 5.39e57T^{2} \) |
| 5 | \( 1 + 2.23e41T + 3.76e84T^{2} \) |
| 7 | \( 1 - 1.98e51T + 1.80e102T^{2} \) |
| 11 | \( 1 - 1.11e63T + 1.01e126T^{2} \) |
| 13 | \( 1 - 2.72e67T + 6.12e134T^{2} \) |
| 17 | \( 1 + 1.03e74T + 7.66e148T^{2} \) |
| 19 | \( 1 + 2.25e77T + 5.36e154T^{2} \) |
| 23 | \( 1 + 1.89e82T + 5.87e164T^{2} \) |
| 29 | \( 1 + 2.78e88T + 8.91e176T^{2} \) |
| 31 | \( 1 + 1.00e90T + 2.84e180T^{2} \) |
| 37 | \( 1 + 1.04e95T + 5.65e189T^{2} \) |
| 41 | \( 1 + 3.09e96T + 1.40e195T^{2} \) |
| 43 | \( 1 - 3.87e98T + 4.46e197T^{2} \) |
| 47 | \( 1 + 1.16e101T + 2.10e202T^{2} \) |
| 53 | \( 1 + 2.20e104T + 4.33e208T^{2} \) |
| 59 | \( 1 + 1.62e107T + 1.87e214T^{2} \) |
| 61 | \( 1 + 1.71e107T + 1.05e216T^{2} \) |
| 67 | \( 1 + 1.04e110T + 9.01e220T^{2} \) |
| 71 | \( 1 + 1.25e112T + 1.00e224T^{2} \) |
| 73 | \( 1 - 8.19e112T + 2.89e225T^{2} \) |
| 79 | \( 1 - 4.20e114T + 4.10e229T^{2} \) |
| 83 | \( 1 + 6.56e115T + 1.61e232T^{2} \) |
| 89 | \( 1 - 6.88e117T + 7.51e235T^{2} \) |
| 97 | \( 1 + 2.55e119T + 2.50e240T^{2} \) |
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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
−11.66861856520640629211357542431, −10.86001494555245995881609417001, −9.186093742829260229609301829007, −8.477789238773626302074646165463, −7.68855864808498488552701974501, −6.16320185498948196461870411752, −3.67180130888566367765870692619, −1.98784233832501905113234329063, −1.54655743489228463980833212194, 0,
1.54655743489228463980833212194, 1.98784233832501905113234329063, 3.67180130888566367765870692619, 6.16320185498948196461870411752, 7.68855864808498488552701974501, 8.477789238773626302074646165463, 9.186093742829260229609301829007, 10.86001494555245995881609417001, 11.66861856520640629211357542431
