| L(s) = 1 | − 4.25e17·2-s − 2.69e28·3-s − 4.83e35·4-s − 2.34e41·5-s + 1.14e46·6-s + 5.11e49·7-s + 4.88e53·8-s + 1.29e56·9-s + 9.97e58·10-s + 4.16e61·11-s + 1.30e64·12-s + 4.22e65·13-s − 2.17e67·14-s + 6.32e69·15-s + 1.13e71·16-s − 2.39e72·17-s − 5.50e73·18-s − 1.78e76·19-s + 1.13e77·20-s − 1.38e78·21-s − 1.77e79·22-s − 1.12e80·23-s − 1.31e82·24-s − 9.55e82·25-s − 1.80e83·26-s + 1.26e85·27-s − 2.47e85·28-s + ⋯ |
| L(s) = 1 | − 0.522·2-s − 1.10·3-s − 0.727·4-s − 0.604·5-s + 0.575·6-s + 0.266·7-s + 0.901·8-s + 0.215·9-s + 0.315·10-s + 0.453·11-s + 0.802·12-s + 0.222·13-s − 0.139·14-s + 0.666·15-s + 0.256·16-s − 0.146·17-s − 0.112·18-s − 1.46·19-s + 0.439·20-s − 0.293·21-s − 0.236·22-s − 0.106·23-s − 0.994·24-s − 0.634·25-s − 0.115·26-s + 0.864·27-s − 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(120-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+59.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(60)\) |
\(\approx\) |
\(0.1751302702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1751302702\) |
| \(L(\frac{121}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 4.25e17T + 6.64e35T^{2} \) |
| 3 | \( 1 + 2.69e28T + 5.99e56T^{2} \) |
| 5 | \( 1 + 2.34e41T + 1.50e83T^{2} \) |
| 7 | \( 1 - 5.11e49T + 3.68e100T^{2} \) |
| 11 | \( 1 - 4.16e61T + 8.42e123T^{2} \) |
| 13 | \( 1 - 4.22e65T + 3.62e132T^{2} \) |
| 17 | \( 1 + 2.39e72T + 2.65e146T^{2} \) |
| 19 | \( 1 + 1.78e76T + 1.48e152T^{2} \) |
| 23 | \( 1 + 1.12e80T + 1.11e162T^{2} \) |
| 29 | \( 1 + 8.31e86T + 1.06e174T^{2} \) |
| 31 | \( 1 + 7.36e88T + 2.96e177T^{2} \) |
| 37 | \( 1 - 2.39e93T + 4.13e186T^{2} \) |
| 41 | \( 1 + 1.33e96T + 8.34e191T^{2} \) |
| 43 | \( 1 - 1.29e97T + 2.41e194T^{2} \) |
| 47 | \( 1 + 4.60e99T + 9.54e198T^{2} \) |
| 53 | \( 1 - 2.14e102T + 1.54e205T^{2} \) |
| 59 | \( 1 + 2.51e105T + 5.38e210T^{2} \) |
| 61 | \( 1 + 1.71e106T + 2.84e212T^{2} \) |
| 67 | \( 1 + 3.54e107T + 2.00e217T^{2} \) |
| 71 | \( 1 + 3.54e109T + 1.99e220T^{2} \) |
| 73 | \( 1 + 5.96e110T + 5.43e221T^{2} \) |
| 79 | \( 1 + 2.77e112T + 6.57e225T^{2} \) |
| 83 | \( 1 - 1.80e114T + 2.34e228T^{2} \) |
| 89 | \( 1 - 1.48e116T + 9.49e231T^{2} \) |
| 97 | \( 1 - 2.63e118T + 2.66e236T^{2} \) |
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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
−12.99556007741004013544361435420, −11.59650038518542832201546513315, −10.59535277044066613159119924841, −9.036490724597526414993759590760, −7.81454973434282697526352396537, −6.22827085902327875083042638385, −4.88902061702803865888019155079, −3.84766763335233493721312745920, −1.60087951802738100997925846912, −0.24585296323749857235798513371,
0.24585296323749857235798513371, 1.60087951802738100997925846912, 3.84766763335233493721312745920, 4.88902061702803865888019155079, 6.22827085902327875083042638385, 7.81454973434282697526352396537, 9.036490724597526414993759590760, 10.59535277044066613159119924841, 11.59650038518542832201546513315, 12.99556007741004013544361435420
