| L(s) = 1 | − 1.44e11·2-s − 1.69e22·4-s − 2.13e26·5-s + 5.63e30·7-s + 7.89e33·8-s + 3.08e37·10-s − 2.12e39·11-s − 7.22e41·13-s − 8.12e41·14-s − 4.98e44·16-s − 6.52e45·17-s + 4.29e46·19-s + 3.62e48·20-s + 3.06e50·22-s − 9.35e50·23-s + 1.91e52·25-s + 1.04e53·26-s − 9.55e52·28-s + 1.21e55·29-s + 7.18e55·31-s − 2.26e56·32-s + 9.40e56·34-s − 1.20e57·35-s − 8.71e58·37-s − 6.19e57·38-s − 1.68e60·40-s − 8.48e59·41-s + ⋯ |
| L(s) = 1 | − 0.742·2-s − 0.449·4-s − 1.31·5-s + 0.114·7-s + 1.07·8-s + 0.974·10-s − 1.88·11-s − 1.21·13-s − 0.0851·14-s − 0.349·16-s − 0.470·17-s + 0.0478·19-s + 0.589·20-s + 1.39·22-s − 0.805·23-s + 0.723·25-s + 0.904·26-s − 0.0515·28-s + 1.76·29-s + 0.851·31-s − 0.816·32-s + 0.349·34-s − 0.150·35-s − 1.35·37-s − 0.0355·38-s − 1.41·40-s − 0.281·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.44e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 2.13e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 5.63e30T + 2.41e63T^{2} \) |
| 11 | \( 1 + 2.12e39T + 1.27e78T^{2} \) |
| 13 | \( 1 + 7.22e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 6.52e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 4.29e46T + 8.06e95T^{2} \) |
| 23 | \( 1 + 9.35e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 1.21e55T + 4.78e109T^{2} \) |
| 31 | \( 1 - 7.18e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 8.71e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 8.48e59T + 9.09e120T^{2} \) |
| 43 | \( 1 + 5.45e60T + 3.23e122T^{2} \) |
| 47 | \( 1 + 7.91e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 2.72e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 3.72e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.27e67T + 7.93e133T^{2} \) |
| 67 | \( 1 - 1.96e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 1.45e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 8.19e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 2.17e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 9.83e71T + 8.52e143T^{2} \) |
| 89 | \( 1 - 6.97e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 5.63e74T + 1.01e149T^{2} \) |
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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
−9.860548469776364748758309840500, −8.252046711762948684650300737028, −8.048352293676148983520758667197, −6.97806848008952373471931025064, −5.05339088000410774087101603881, −4.55681708131087496698538314528, −3.22778172318570535328470302373, −2.09975643263715045446159796793, −0.56991641342663100348240340225, 0,
0.56991641342663100348240340225, 2.09975643263715045446159796793, 3.22778172318570535328470302373, 4.55681708131087496698538314528, 5.05339088000410774087101603881, 6.97806848008952373471931025064, 8.048352293676148983520758667197, 8.252046711762948684650300737028, 9.860548469776364748758309840500
