| L(s) = 1 | − 2.24e13·2-s − 6.44e20·3-s − 1.14e26·4-s − 1.81e31·5-s + 1.44e34·6-s − 2.76e37·7-s + 1.64e40·8-s − 2.49e42·9-s + 4.07e44·10-s + 2.89e46·11-s + 7.37e46·12-s − 2.93e49·13-s + 6.20e50·14-s + 1.16e52·15-s − 2.99e53·16-s + 3.53e54·17-s + 5.60e55·18-s − 2.73e56·19-s + 2.07e57·20-s + 1.78e58·21-s − 6.49e59·22-s + 5.99e60·23-s − 1.06e61·24-s + 1.66e62·25-s + 6.58e62·26-s + 3.48e63·27-s + 3.15e63·28-s + ⋯ |
| L(s) = 1 | − 0.902·2-s − 0.378·3-s − 0.184·4-s − 1.42·5-s + 0.341·6-s − 0.682·7-s + 1.06·8-s − 0.857·9-s + 1.28·10-s + 1.31·11-s + 0.0698·12-s − 0.787·13-s + 0.616·14-s + 0.538·15-s − 0.781·16-s + 0.621·17-s + 0.773·18-s − 0.340·19-s + 0.263·20-s + 0.258·21-s − 1.18·22-s + 1.51·23-s − 0.404·24-s + 1.03·25-s + 0.711·26-s + 0.702·27-s + 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(45)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{91}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.24e13T + 6.18e26T^{2} \) |
| 3 | \( 1 + 6.44e20T + 2.90e42T^{2} \) |
| 5 | \( 1 + 1.81e31T + 1.61e62T^{2} \) |
| 7 | \( 1 + 2.76e37T + 1.63e75T^{2} \) |
| 11 | \( 1 - 2.89e46T + 4.83e92T^{2} \) |
| 13 | \( 1 + 2.93e49T + 1.38e99T^{2} \) |
| 17 | \( 1 - 3.53e54T + 3.23e109T^{2} \) |
| 19 | \( 1 + 2.73e56T + 6.44e113T^{2} \) |
| 23 | \( 1 - 5.99e60T + 1.56e121T^{2} \) |
| 29 | \( 1 - 2.05e65T + 1.42e130T^{2} \) |
| 31 | \( 1 + 1.59e66T + 5.38e132T^{2} \) |
| 37 | \( 1 - 1.13e69T + 3.71e139T^{2} \) |
| 41 | \( 1 - 8.14e70T + 3.44e143T^{2} \) |
| 43 | \( 1 - 3.18e72T + 2.39e145T^{2} \) |
| 47 | \( 1 + 3.19e74T + 6.55e148T^{2} \) |
| 53 | \( 1 + 1.04e77T + 2.88e153T^{2} \) |
| 59 | \( 1 - 8.65e78T + 4.03e157T^{2} \) |
| 61 | \( 1 + 3.87e79T + 7.84e158T^{2} \) |
| 67 | \( 1 - 1.41e81T + 3.31e162T^{2} \) |
| 71 | \( 1 - 1.52e82T + 5.78e164T^{2} \) |
| 73 | \( 1 + 5.28e82T + 6.85e165T^{2} \) |
| 79 | \( 1 - 4.95e84T + 7.74e168T^{2} \) |
| 83 | \( 1 - 5.30e84T + 6.27e170T^{2} \) |
| 89 | \( 1 + 6.27e86T + 3.13e173T^{2} \) |
| 97 | \( 1 - 2.15e88T + 6.64e176T^{2} \) |
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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
−14.53074258366583036640821778686, −12.34084534606860609110999755448, −11.09969066201492674814066400277, −9.394939964391648639794585083696, −8.182079039762662653481847604366, −6.79563336319194099738029645743, −4.68605182707309679630770282207, −3.28763863480721116545096077530, −0.917234848080778442013681445998, 0,
0.917234848080778442013681445998, 3.28763863480721116545096077530, 4.68605182707309679630770282207, 6.79563336319194099738029645743, 8.182079039762662653481847604366, 9.394939964391648639794585083696, 11.09969066201492674814066400277, 12.34084534606860609110999755448, 14.53074258366583036640821778686
