| L(s) = 1 | + 0.130·2-s − 3-s − 1.98·4-s + 3.48·5-s − 0.130·6-s − 7-s − 0.518·8-s + 9-s + 0.454·10-s − 2.86·11-s + 1.98·12-s − 5.71·13-s − 0.130·14-s − 3.48·15-s + 3.89·16-s − 1.00·17-s + 0.130·18-s − 6.18·19-s − 6.91·20-s + 21-s − 0.373·22-s − 4.54·23-s + 0.518·24-s + 7.16·25-s − 0.744·26-s − 27-s + 1.98·28-s + ⋯ |
| L(s) = 1 | + 0.0920·2-s − 0.577·3-s − 0.991·4-s + 1.56·5-s − 0.0531·6-s − 0.377·7-s − 0.183·8-s + 0.333·9-s + 0.143·10-s − 0.864·11-s + 0.572·12-s − 1.58·13-s − 0.0347·14-s − 0.900·15-s + 0.974·16-s − 0.243·17-s + 0.0306·18-s − 1.42·19-s − 1.54·20-s + 0.218·21-s − 0.0795·22-s − 0.948·23-s + 0.105·24-s + 1.43·25-s − 0.145·26-s − 0.192·27-s + 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7011957987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7011957987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.130T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + 1.00T + 17T^{2} \) |
| 19 | \( 1 + 6.18T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 - 3.97T + 37T^{2} \) |
| 41 | \( 1 - 0.368T + 41T^{2} \) |
| 43 | \( 1 + 0.179T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 5.49T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 1.49T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 1.43T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.80836686404424919999561496875, −7.03196706059143045928502222907, −6.22075764118512111455804904508, −5.63348065749806296073530153766, −5.18236242983062216512751636541, −4.52447758141218536889307695493, −3.63675636682037754360378891836, −2.38039151888155582179123164388, −1.97503076730746744912456573760, −0.39869634116077572889366275039,
0.39869634116077572889366275039, 1.97503076730746744912456573760, 2.38039151888155582179123164388, 3.63675636682037754360378891836, 4.52447758141218536889307695493, 5.18236242983062216512751636541, 5.63348065749806296073530153766, 6.22075764118512111455804904508, 7.03196706059143045928502222907, 7.80836686404424919999561496875
