| L(s) = 1 | − 1.61·2-s + 0.110·3-s + 0.592·4-s − 0.517·5-s − 0.177·6-s − 3.40·7-s + 2.26·8-s − 2.98·9-s + 0.833·10-s + 4.35·11-s + 0.0653·12-s + 13-s + 5.48·14-s − 0.0570·15-s − 4.83·16-s + 4.44·17-s + 4.81·18-s + 4.53·19-s − 0.306·20-s − 0.375·21-s − 7.01·22-s + 3.32·23-s + 0.249·24-s − 4.73·25-s − 1.61·26-s − 0.660·27-s − 2.01·28-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.0636·3-s + 0.296·4-s − 0.231·5-s − 0.0724·6-s − 1.28·7-s + 0.801·8-s − 0.995·9-s + 0.263·10-s + 1.31·11-s + 0.0188·12-s + 0.277·13-s + 1.46·14-s − 0.0147·15-s − 1.20·16-s + 1.07·17-s + 1.13·18-s + 1.04·19-s − 0.0685·20-s − 0.0820·21-s − 1.49·22-s + 0.693·23-s + 0.0510·24-s − 0.946·25-s − 0.315·26-s − 0.127·27-s − 0.381·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.110T + 3T^{2} \) |
| 5 | \( 1 + 0.517T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 23 | \( 1 - 3.32T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 + 8.15T + 31T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 - 0.0385T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 + 9.85T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 + 0.134T + 73T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−9.402886887672651354862177343978, −9.007187473246081294697990485321, −7.952936729141334614280492091408, −7.25614793526745490958184180481, −6.28720827262608754797934217339, −5.40766899774360930575928444649, −3.86138523441167285928082749190, −3.15822413245637700747838273886, −1.42740835386902538296936637496, 0,
1.42740835386902538296936637496, 3.15822413245637700747838273886, 3.86138523441167285928082749190, 5.40766899774360930575928444649, 6.28720827262608754797934217339, 7.25614793526745490958184180481, 7.952936729141334614280492091408, 9.007187473246081294697990485321, 9.402886887672651354862177343978
