L(s) = 1 | + 5.53e8·2-s − 2.70e13·3-s + 1.62e17·4-s − 5.73e19·5-s − 1.49e22·6-s + 1.73e24·7-s + 1.00e25·8-s − 8.39e26·9-s − 3.17e28·10-s − 7.30e29·11-s − 4.38e30·12-s − 7.35e31·13-s + 9.58e32·14-s + 1.55e33·15-s − 1.78e34·16-s − 5.76e34·17-s − 4.64e35·18-s + 2.98e36·19-s − 9.31e36·20-s − 4.68e37·21-s − 4.04e38·22-s − 2.19e38·23-s − 2.71e38·24-s − 3.64e39·25-s − 4.07e40·26-s + 6.51e40·27-s + 2.81e41·28-s + ⋯ |
L(s) = 1 | + 1.45·2-s − 0.682·3-s + 1.12·4-s − 0.688·5-s − 0.994·6-s + 1.42·7-s + 0.183·8-s − 0.534·9-s − 1.00·10-s − 1.52·11-s − 0.768·12-s − 1.31·13-s + 2.07·14-s + 0.469·15-s − 0.857·16-s − 0.493·17-s − 0.779·18-s + 1.07·19-s − 0.775·20-s − 0.970·21-s − 2.22·22-s − 0.340·23-s − 0.125·24-s − 0.525·25-s − 1.91·26-s + 1.04·27-s + 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(58-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(29)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{59}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 - 5.53e8T + 1.44e17T^{2} \) |
| 3 | \( 1 + 2.70e13T + 1.57e27T^{2} \) |
| 5 | \( 1 + 5.73e19T + 6.93e39T^{2} \) |
| 7 | \( 1 - 1.73e24T + 1.48e48T^{2} \) |
| 11 | \( 1 + 7.30e29T + 2.28e59T^{2} \) |
| 13 | \( 1 + 7.35e31T + 3.12e63T^{2} \) |
| 17 | \( 1 + 5.76e34T + 1.36e70T^{2} \) |
| 19 | \( 1 - 2.98e36T + 7.74e72T^{2} \) |
| 23 | \( 1 + 2.19e38T + 4.15e77T^{2} \) |
| 29 | \( 1 - 2.03e41T + 2.27e83T^{2} \) |
| 31 | \( 1 + 2.38e41T + 1.01e85T^{2} \) |
| 37 | \( 1 - 2.29e44T + 2.44e89T^{2} \) |
| 41 | \( 1 + 8.95e45T + 8.48e91T^{2} \) |
| 43 | \( 1 - 7.06e46T + 1.28e93T^{2} \) |
| 47 | \( 1 - 3.69e47T + 2.03e95T^{2} \) |
| 53 | \( 1 + 2.42e49T + 1.92e98T^{2} \) |
| 59 | \( 1 + 8.00e49T + 8.68e100T^{2} \) |
| 61 | \( 1 + 4.03e50T + 5.80e101T^{2} \) |
| 67 | \( 1 - 9.38e51T + 1.21e104T^{2} \) |
| 71 | \( 1 + 5.52e51T + 3.32e105T^{2} \) |
| 73 | \( 1 + 6.25e52T + 1.61e106T^{2} \) |
| 79 | \( 1 - 2.30e53T + 1.46e108T^{2} \) |
| 83 | \( 1 + 1.95e54T + 2.44e109T^{2} \) |
| 89 | \( 1 - 2.14e54T + 1.30e111T^{2} \) |
| 97 | \( 1 + 7.81e56T + 1.76e113T^{2} \) |
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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
−17.78398667890156681143724554123, −15.57298211810975379825866067611, −14.15143853936896490345321812589, −12.20477665507040089983111728082, −11.15650724356567680291578643487, −7.72492792130318511772563790365, −5.44652409327183811555795234813, −4.63143830175394192625812911713, −2.60299345341729400408032905577, 0,
2.60299345341729400408032905577, 4.63143830175394192625812911713, 5.44652409327183811555795234813, 7.72492792130318511772563790365, 11.15650724356567680291578643487, 12.20477665507040089983111728082, 14.15143853936896490345321812589, 15.57298211810975379825866067611, 17.78398667890156681143724554123