| L(s) = 1 | − 4.84·2-s − 0.335·3-s + 15.5·4-s + 5·5-s + 1.62·6-s − 24.3·7-s − 36.4·8-s − 26.8·9-s − 24.2·10-s − 11·11-s − 5.20·12-s − 65.8·13-s + 118.·14-s − 1.67·15-s + 52.5·16-s + 107.·17-s + 130.·18-s − 19·19-s + 77.5·20-s + 8.16·21-s + 53.3·22-s + 155.·23-s + 12.2·24-s + 25·25-s + 319.·26-s + 18.0·27-s − 377.·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.0645·3-s + 1.93·4-s + 0.447·5-s + 0.110·6-s − 1.31·7-s − 1.60·8-s − 0.995·9-s − 0.766·10-s − 0.301·11-s − 0.125·12-s − 1.40·13-s + 2.25·14-s − 0.0288·15-s + 0.820·16-s + 1.53·17-s + 1.70·18-s − 0.229·19-s + 0.867·20-s + 0.0847·21-s + 0.516·22-s + 1.41·23-s + 0.103·24-s + 0.200·25-s + 2.40·26-s + 0.128·27-s − 2.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 4.84T + 8T^{2} \) |
| 3 | \( 1 + 0.335T + 27T^{2} \) |
| 7 | \( 1 + 24.3T + 343T^{2} \) |
| 13 | \( 1 + 65.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 155.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 261.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 84.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 289.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 411.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 255.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 725.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 23.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 605.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 458.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 76.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 226.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 810.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 619.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 741.T + 9.12e5T^{2} \) |
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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
−9.312625659731595676636912453419, −8.457233577729596734291281758298, −7.61007593642816519300810836510, −6.82142751765550494220394084097, −6.06373471001024500575142232417, −5.03624017576347247908591416657, −3.02984570403028751891604708229, −2.59839425187225890553221369158, −0.985059364181577858828849339710, 0,
0.985059364181577858828849339710, 2.59839425187225890553221369158, 3.02984570403028751891604708229, 5.03624017576347247908591416657, 6.06373471001024500575142232417, 6.82142751765550494220394084097, 7.61007593642816519300810836510, 8.457233577729596734291281758298, 9.312625659731595676636912453419
