| L(s) = 1 | − 1.09e16·2-s − 5.69e25·3-s − 4.17e31·4-s + 2.81e36·5-s + 6.25e41·6-s + 3.42e44·7-s + 2.23e48·8-s + 2.11e51·9-s − 3.09e52·10-s + 2.99e55·11-s + 2.37e57·12-s + 6.92e59·13-s − 3.75e60·14-s − 1.60e62·15-s − 1.78e64·16-s + 1.03e66·17-s − 2.32e67·18-s + 7.83e67·19-s − 1.17e68·20-s − 1.95e70·21-s − 3.28e71·22-s + 3.41e72·23-s − 1.27e74·24-s − 6.08e74·25-s − 7.60e75·26-s − 5.65e76·27-s − 1.42e76·28-s + ⋯ |
| L(s) = 1 | − 0.861·2-s − 1.69·3-s − 0.257·4-s + 0.113·5-s + 1.46·6-s + 0.209·7-s + 1.08·8-s + 1.88·9-s − 0.0977·10-s + 0.578·11-s + 0.436·12-s + 1.75·13-s − 0.180·14-s − 0.192·15-s − 0.676·16-s + 1.53·17-s − 1.62·18-s + 0.302·19-s − 0.0291·20-s − 0.356·21-s − 0.498·22-s + 0.479·23-s − 1.83·24-s − 0.987·25-s − 1.51·26-s − 1.49·27-s − 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(54)\) |
\(\approx\) |
\(0.9617777005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9617777005\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.09e16T + 1.62e32T^{2} \) |
| 3 | \( 1 + 5.69e25T + 1.12e51T^{2} \) |
| 5 | \( 1 - 2.81e36T + 6.16e74T^{2} \) |
| 7 | \( 1 - 3.42e44T + 2.66e90T^{2} \) |
| 11 | \( 1 - 2.99e55T + 2.68e111T^{2} \) |
| 13 | \( 1 - 6.92e59T + 1.55e119T^{2} \) |
| 17 | \( 1 - 1.03e66T + 4.55e131T^{2} \) |
| 19 | \( 1 - 7.83e67T + 6.70e136T^{2} \) |
| 23 | \( 1 - 3.41e72T + 5.06e145T^{2} \) |
| 29 | \( 1 - 1.31e78T + 2.99e156T^{2} \) |
| 31 | \( 1 - 1.11e80T + 3.76e159T^{2} \) |
| 37 | \( 1 - 7.06e83T + 6.27e167T^{2} \) |
| 41 | \( 1 + 2.42e86T + 3.69e172T^{2} \) |
| 43 | \( 1 - 3.45e87T + 6.04e174T^{2} \) |
| 47 | \( 1 - 1.83e89T + 8.21e178T^{2} \) |
| 53 | \( 1 - 1.73e92T + 3.14e184T^{2} \) |
| 59 | \( 1 - 1.65e94T + 3.02e189T^{2} \) |
| 61 | \( 1 - 2.05e95T + 1.07e191T^{2} \) |
| 67 | \( 1 + 8.20e96T + 2.45e195T^{2} \) |
| 71 | \( 1 - 1.83e99T + 1.21e198T^{2} \) |
| 73 | \( 1 - 2.72e99T + 2.37e199T^{2} \) |
| 79 | \( 1 - 2.64e101T + 1.11e203T^{2} \) |
| 83 | \( 1 - 5.13e102T + 2.19e205T^{2} \) |
| 89 | \( 1 + 1.30e104T + 3.84e208T^{2} \) |
| 97 | \( 1 - 2.53e105T + 3.84e212T^{2} \) |
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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
−13.64684317144811548147469679188, −11.97772384600680331405663227204, −10.84687802753710558499792851612, −9.761572475484831913371101744113, −8.113007816636755105169670602011, −6.47911112102385704004207512019, −5.35536715326980745285247894267, −4.00807649857750845239722251941, −1.18690891918065881811111971990, −0.837147607785070088334852904639,
0.837147607785070088334852904639, 1.18690891918065881811111971990, 4.00807649857750845239722251941, 5.35536715326980745285247894267, 6.47911112102385704004207512019, 8.113007816636755105169670602011, 9.761572475484831913371101744113, 10.84687802753710558499792851612, 11.97772384600680331405663227204, 13.64684317144811548147469679188
