| L(s) = 1 | − 9.60e10·2-s − 2.85e22·4-s + 1.49e26·5-s − 3.16e31·7-s + 6.37e33·8-s − 1.43e37·10-s + 9.04e38·11-s − 1.02e42·13-s + 3.03e42·14-s + 4.66e44·16-s − 2.82e45·17-s + 5.13e47·19-s − 4.27e48·20-s − 8.68e49·22-s + 1.06e51·23-s − 4.07e51·25-s + 9.83e52·26-s + 9.02e53·28-s + 5.54e54·29-s − 6.85e55·31-s − 2.85e56·32-s + 2.71e56·34-s − 4.73e57·35-s + 1.00e59·37-s − 4.93e58·38-s + 9.53e59·40-s − 4.31e60·41-s + ⋯ |
| L(s) = 1 | − 0.494·2-s − 0.755·4-s + 0.919·5-s − 0.643·7-s + 0.867·8-s − 0.454·10-s + 0.801·11-s − 1.72·13-s + 0.318·14-s + 0.326·16-s − 0.203·17-s + 0.571·19-s − 0.695·20-s − 0.396·22-s + 0.920·23-s − 0.153·25-s + 0.853·26-s + 0.486·28-s + 0.801·29-s − 0.812·31-s − 1.02·32-s + 0.100·34-s − 0.592·35-s + 1.55·37-s − 0.282·38-s + 0.798·40-s − 1.43·41-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 | \( 1 + 9.60e10T + 3.77e22T^{2} \) |
| 5 | \( 1 - 1.49e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 3.16e31T + 2.41e63T^{2} \) |
| 11 | \( 1 - 9.04e38T + 1.27e78T^{2} \) |
| 13 | \( 1 + 1.02e42T + 3.51e83T^{2} \) |
| 17 | \( 1 + 2.82e45T + 1.92e92T^{2} \) |
| 19 | \( 1 - 5.13e47T + 8.06e95T^{2} \) |
| 23 | \( 1 - 1.06e51T + 1.34e102T^{2} \) |
| 29 | \( 1 - 5.54e54T + 4.78e109T^{2} \) |
| 31 | \( 1 + 6.85e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.00e59T + 4.12e117T^{2} \) |
| 41 | \( 1 + 4.31e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 1.74e61T + 3.23e122T^{2} \) |
| 47 | \( 1 - 9.73e62T + 2.55e125T^{2} \) |
| 53 | \( 1 - 2.55e63T + 2.09e129T^{2} \) |
| 59 | \( 1 + 5.98e65T + 6.51e132T^{2} \) |
| 61 | \( 1 + 1.23e67T + 7.93e133T^{2} \) |
| 67 | \( 1 + 9.36e67T + 9.02e136T^{2} \) |
| 71 | \( 1 - 3.34e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 1.44e70T + 5.61e139T^{2} \) |
| 79 | \( 1 - 1.89e71T + 2.09e142T^{2} \) |
| 83 | \( 1 - 1.25e72T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.25e73T + 1.60e146T^{2} \) |
| 97 | \( 1 + 8.39e73T + 1.01e149T^{2} \) |
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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
−9.557449876377533522521693534450, −9.111039472531314977684350685891, −7.64336762481290484457851994823, −6.60432288555987192614140744541, −5.37310263897879659859008335174, −4.49488802567913280587814576359, −3.17500128881955722041128260096, −2.02486725830125713365423255948, −0.965956623043206634757132764099, 0,
0.965956623043206634757132764099, 2.02486725830125713365423255948, 3.17500128881955722041128260096, 4.49488802567913280587814576359, 5.37310263897879659859008335174, 6.60432288555987192614140744541, 7.64336762481290484457851994823, 9.111039472531314977684350685891, 9.557449876377533522521693534450
