| L(s) = 1 | + 0.313·2-s − 1.90·4-s − 0.0595·5-s + 2.67·7-s − 1.22·8-s − 0.0186·10-s − 5.42·11-s − 0.468·13-s + 0.838·14-s + 3.42·16-s − 6.86·19-s + 0.113·20-s − 1.69·22-s − 5.26·23-s − 4.99·25-s − 0.146·26-s − 5.09·28-s + 2.40·29-s + 3.15·31-s + 3.51·32-s − 0.159·35-s − 7.50·37-s − 2.15·38-s + 0.0727·40-s + 2.22·41-s − 4.51·43-s + 10.3·44-s + ⋯ |
| L(s) = 1 | + 0.221·2-s − 0.950·4-s − 0.0266·5-s + 1.01·7-s − 0.431·8-s − 0.00589·10-s − 1.63·11-s − 0.130·13-s + 0.224·14-s + 0.855·16-s − 1.57·19-s + 0.0253·20-s − 0.362·22-s − 1.09·23-s − 0.999·25-s − 0.0288·26-s − 0.962·28-s + 0.447·29-s + 0.566·31-s + 0.621·32-s − 0.0269·35-s − 1.23·37-s − 0.348·38-s + 0.0115·40-s + 0.347·41-s − 0.688·43-s + 1.55·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038483484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038483484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.313T + 2T^{2} \) |
| 5 | \( 1 + 0.0595T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + 0.468T + 13T^{2} \) |
| 19 | \( 1 + 6.86T + 19T^{2} \) |
| 23 | \( 1 + 5.26T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 + 7.50T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 1.82T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 8.70T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−8.142933053171055231802363650567, −7.35612043325868856214645208569, −6.30619181917473097920154492781, −5.51561433987261260205400741106, −5.08297045790138876241577369860, −4.32990570017933202381504096637, −3.79409611826157342719387655865, −2.58643037431270574768619240635, −1.92366732517957681715766893406, −0.47538235412164115585251731979,
0.47538235412164115585251731979, 1.92366732517957681715766893406, 2.58643037431270574768619240635, 3.79409611826157342719387655865, 4.32990570017933202381504096637, 5.08297045790138876241577369860, 5.51561433987261260205400741106, 6.30619181917473097920154492781, 7.35612043325868856214645208569, 8.142933053171055231802363650567
