| L(s) = 1 | + 2.14e11·2-s + 8.19e21·4-s − 3.21e24·5-s + 4.19e31·7-s − 6.34e33·8-s − 6.89e35·10-s + 2.92e38·11-s − 4.16e41·13-s + 8.99e42·14-s − 1.66e45·16-s − 8.08e45·17-s + 9.18e47·19-s − 2.63e46·20-s + 6.26e49·22-s + 1.36e51·23-s − 2.64e52·25-s − 8.92e52·26-s + 3.43e53·28-s + 7.92e54·29-s − 6.99e54·31-s − 1.18e56·32-s − 1.73e57·34-s − 1.34e56·35-s + 1.39e58·37-s + 1.96e59·38-s + 2.03e58·40-s + 6.26e58·41-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.216·4-s − 0.0197·5-s + 0.854·7-s − 0.863·8-s − 0.0217·10-s + 0.258·11-s − 0.702·13-s + 0.942·14-s − 1.16·16-s − 0.582·17-s + 1.02·19-s − 0.00428·20-s + 0.285·22-s + 1.17·23-s − 0.999·25-s − 0.774·26-s + 0.185·28-s + 1.14·29-s − 0.0829·31-s − 0.426·32-s − 0.643·34-s − 0.0168·35-s + 0.217·37-s + 1.12·38-s + 0.0170·40-s + 0.0207·41-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 | \( 1 - 2.14e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 3.21e24T + 2.64e52T^{2} \) |
| 7 | \( 1 - 4.19e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 2.92e38T + 1.27e78T^{2} \) |
| 13 | \( 1 + 4.16e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 8.08e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 9.18e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 1.36e51T + 1.34e102T^{2} \) |
| 29 | \( 1 - 7.92e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 6.99e54T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.39e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 6.26e58T + 9.09e120T^{2} \) |
| 43 | \( 1 + 2.68e60T + 3.23e122T^{2} \) |
| 47 | \( 1 + 3.53e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 3.88e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 1.63e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.31e67T + 7.93e133T^{2} \) |
| 67 | \( 1 + 5.25e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 1.21e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 3.72e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 6.93e70T + 2.09e142T^{2} \) |
| 83 | \( 1 + 1.54e72T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.45e73T + 1.60e146T^{2} \) |
| 97 | \( 1 - 2.65e74T + 1.01e149T^{2} \) |
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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
−9.746671553887316555087792679359, −8.663536559876493379258824036518, −7.42971123511552852964329280435, −6.25102888197689430440990126881, −5.08467985975365027974417655747, −4.59258833060196587070740922801, −3.42692676254443525069270792868, −2.46272439035450692045517451994, −1.24489101123771784438717240571, 0,
1.24489101123771784438717240571, 2.46272439035450692045517451994, 3.42692676254443525069270792868, 4.59258833060196587070740922801, 5.08467985975365027974417655747, 6.25102888197689430440990126881, 7.42971123511552852964329280435, 8.663536559876493379258824036518, 9.746671553887316555087792679359
