| L(s) = 1 | − 1.80e11·2-s − 5.18e21·4-s − 2.29e26·5-s + 5.15e31·7-s + 7.75e33·8-s + 4.14e37·10-s + 1.35e39·11-s − 4.62e41·13-s − 9.29e42·14-s − 1.20e45·16-s + 2.53e46·17-s − 8.62e47·19-s + 1.19e48·20-s − 2.44e50·22-s + 2.00e51·23-s + 2.62e52·25-s + 8.34e52·26-s − 2.67e53·28-s − 9.19e54·29-s − 9.56e55·31-s − 7.55e55·32-s − 4.56e57·34-s − 1.18e58·35-s + 2.28e58·37-s + 1.55e59·38-s − 1.78e60·40-s − 4.33e59·41-s + ⋯ |
| L(s) = 1 | − 0.928·2-s − 0.137·4-s − 1.41·5-s + 1.04·7-s + 1.05·8-s + 1.31·10-s + 1.20·11-s − 0.779·13-s − 0.974·14-s − 0.843·16-s + 1.82·17-s − 0.960·19-s + 0.193·20-s − 1.11·22-s + 1.72·23-s + 0.993·25-s + 0.724·26-s − 0.143·28-s − 1.32·29-s − 1.13·31-s − 0.272·32-s − 1.69·34-s − 1.48·35-s + 0.355·37-s + 0.892·38-s − 1.49·40-s − 0.143·41-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 | \( 1 + 1.80e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 2.29e26T + 2.64e52T^{2} \) |
| 7 | \( 1 - 5.15e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.35e39T + 1.27e78T^{2} \) |
| 13 | \( 1 + 4.62e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 2.53e46T + 1.92e92T^{2} \) |
| 19 | \( 1 + 8.62e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 2.00e51T + 1.34e102T^{2} \) |
| 29 | \( 1 + 9.19e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 9.56e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 2.28e58T + 4.12e117T^{2} \) |
| 41 | \( 1 + 4.33e59T + 9.09e120T^{2} \) |
| 43 | \( 1 + 3.06e59T + 3.23e122T^{2} \) |
| 47 | \( 1 + 6.44e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 4.58e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 7.35e65T + 6.51e132T^{2} \) |
| 61 | \( 1 + 1.46e67T + 7.93e133T^{2} \) |
| 67 | \( 1 - 3.08e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 1.57e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 9.34e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 3.85e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 1.24e72T + 8.52e143T^{2} \) |
| 89 | \( 1 + 4.57e72T + 1.60e146T^{2} \) |
| 97 | \( 1 - 1.09e74T + 1.01e149T^{2} \) |
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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
−9.544172232716456694670400266784, −8.591617733405949096027064926317, −7.74454466031610395179678117952, −7.13794605067672204400612142787, −5.20668272505168751470967659121, −4.30249001977717569160463888891, −3.44262450132213755178566092327, −1.70556221271634721710328193858, −0.939651499714252214110468397397, 0,
0.939651499714252214110468397397, 1.70556221271634721710328193858, 3.44262450132213755178566092327, 4.30249001977717569160463888891, 5.20668272505168751470967659121, 7.13794605067672204400612142787, 7.74454466031610395179678117952, 8.591617733405949096027064926317, 9.544172232716456694670400266784
