| L(s) = 1 | − 3.25e11·2-s + 6.79e22·4-s + 8.28e25·5-s − 3.35e31·7-s − 9.79e33·8-s − 2.69e37·10-s − 8.87e37·11-s − 6.04e41·13-s + 1.09e43·14-s + 6.20e44·16-s − 1.93e45·17-s − 8.13e47·19-s + 5.62e48·20-s + 2.88e49·22-s − 1.54e51·23-s − 1.96e52·25-s + 1.96e53·26-s − 2.27e54·28-s + 1.34e55·29-s − 2.44e55·31-s + 1.68e56·32-s + 6.30e56·34-s − 2.77e57·35-s + 6.42e58·37-s + 2.64e59·38-s − 8.11e59·40-s + 2.40e60·41-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.79·4-s + 0.509·5-s − 0.683·7-s − 1.33·8-s − 0.851·10-s − 0.0787·11-s − 1.01·13-s + 1.14·14-s + 0.434·16-s − 0.139·17-s − 0.905·19-s + 0.915·20-s + 0.131·22-s − 1.33·23-s − 0.740·25-s + 1.70·26-s − 1.22·28-s + 1.94·29-s − 0.290·31-s + 0.607·32-s + 0.234·34-s − 0.347·35-s + 1.00·37-s + 1.51·38-s − 0.679·40-s + 0.797·41-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 | \( 1 + 3.25e11T + 3.77e22T^{2} \) |
| 5 | \( 1 - 8.28e25T + 2.64e52T^{2} \) |
| 7 | \( 1 + 3.35e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 8.87e37T + 1.27e78T^{2} \) |
| 13 | \( 1 + 6.04e41T + 3.51e83T^{2} \) |
| 17 | \( 1 + 1.93e45T + 1.92e92T^{2} \) |
| 19 | \( 1 + 8.13e47T + 8.06e95T^{2} \) |
| 23 | \( 1 + 1.54e51T + 1.34e102T^{2} \) |
| 29 | \( 1 - 1.34e55T + 4.78e109T^{2} \) |
| 31 | \( 1 + 2.44e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 6.42e58T + 4.12e117T^{2} \) |
| 41 | \( 1 - 2.40e60T + 9.09e120T^{2} \) |
| 43 | \( 1 - 5.33e60T + 3.23e122T^{2} \) |
| 47 | \( 1 - 4.37e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 2.28e64T + 2.09e129T^{2} \) |
| 59 | \( 1 - 4.09e66T + 6.51e132T^{2} \) |
| 61 | \( 1 - 9.98e66T + 7.93e133T^{2} \) |
| 67 | \( 1 + 2.50e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 2.90e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 1.42e70T + 5.61e139T^{2} \) |
| 79 | \( 1 + 1.34e71T + 2.09e142T^{2} \) |
| 83 | \( 1 + 9.68e71T + 8.52e143T^{2} \) |
| 89 | \( 1 + 1.62e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 2.63e74T + 1.01e149T^{2} \) |
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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
−9.882530851175192658623347808133, −8.800914881834430594584644206383, −7.83092291404878518510249865199, −6.78753704313293703778536804735, −5.91779273070414915184084049366, −4.29267716361790559794783668838, −2.62799085772309962234231494893, −2.06152178057633636843093399911, −0.809506881178551875077393737348, 0,
0.809506881178551875077393737348, 2.06152178057633636843093399911, 2.62799085772309962234231494893, 4.29267716361790559794783668838, 5.91779273070414915184084049366, 6.78753704313293703778536804735, 7.83092291404878518510249865199, 8.800914881834430594584644206383, 9.882530851175192658623347808133
