| L(s) = 1 | + 2.46e11·2-s + 2.28e22·4-s − 3.06e26·5-s − 7.90e31·7-s − 3.68e33·8-s − 7.54e37·10-s + 1.66e39·11-s − 2.77e41·13-s − 1.94e43·14-s − 1.76e45·16-s + 4.78e45·17-s + 1.12e48·19-s − 6.99e48·20-s + 4.09e50·22-s + 6.58e50·23-s + 6.74e52·25-s − 6.83e52·26-s − 1.80e54·28-s + 6.79e54·29-s − 4.78e55·31-s − 2.96e56·32-s + 1.17e57·34-s + 2.42e58·35-s − 2.41e57·37-s + 2.76e59·38-s + 1.12e60·40-s − 3.20e60·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.603·4-s − 1.88·5-s − 1.61·7-s − 0.501·8-s − 2.38·10-s + 1.47·11-s − 0.468·13-s − 2.03·14-s − 1.23·16-s + 0.345·17-s + 1.25·19-s − 1.13·20-s + 1.86·22-s + 0.566·23-s + 2.54·25-s − 0.592·26-s − 0.972·28-s + 0.982·29-s − 0.567·31-s − 1.06·32-s + 0.437·34-s + 3.03·35-s − 0.0375·37-s + 1.58·38-s + 0.944·40-s − 1.06·41-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 | \( 1 - 2.46e11T + 3.77e22T^{2} \) |
| 5 | \( 1 + 3.06e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 7.90e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 1.66e39T + 1.27e78T^{2} \) |
| 13 | \( 1 + 2.77e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 4.78e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.12e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 6.58e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 6.79e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 4.78e55T + 7.11e111T^{2} \) |
| 37 | \( 1 + 2.41e57T + 4.12e117T^{2} \) |
| 41 | \( 1 + 3.20e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 1.93e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 1.44e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 4.38e63T + 2.09e129T^{2} \) |
| 59 | \( 1 - 4.66e66T + 6.51e132T^{2} \) |
| 61 | \( 1 + 7.08e66T + 7.93e133T^{2} \) |
| 67 | \( 1 - 2.44e68T + 9.02e136T^{2} \) |
| 71 | \( 1 - 2.91e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 4.32e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.82e71T + 2.09e142T^{2} \) |
| 83 | \( 1 + 5.34e71T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.36e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 4.35e73T + 1.01e149T^{2} \) |
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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
−9.725365381045433423850347463417, −8.660029375600071885631499482928, −7.12326909647243470255782940262, −6.58496661547370522479804466375, −5.17768321279329319979312365060, −4.08548480229590996995651582732, −3.47975035643741808859008851414, −2.97878856368207707344466533524, −0.882698721159458462960105446745, 0,
0.882698721159458462960105446745, 2.97878856368207707344466533524, 3.47975035643741808859008851414, 4.08548480229590996995651582732, 5.17768321279329319979312365060, 6.58496661547370522479804466375, 7.12326909647243470255782940262, 8.660029375600071885631499482928, 9.725365381045433423850347463417
