| L(s) = 1 | − 2.46·2-s + 4.05·4-s + 2.59·5-s − 5.05·8-s − 6.38·10-s + 4.51·11-s − 13-s + 4.32·16-s + 0.945·17-s + 4.05·19-s + 10.5·20-s − 11.1·22-s − 0.273·23-s + 1.72·25-s + 2.46·26-s + 2.46·29-s − 2.32·31-s − 0.539·32-s − 2.32·34-s + 1.78·37-s − 9.97·38-s − 13.1·40-s + 6.40·41-s − 10.4·43-s + 18.3·44-s + 0.672·46-s + 12.1·47-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.02·4-s + 1.15·5-s − 1.78·8-s − 2.01·10-s + 1.36·11-s − 0.277·13-s + 1.08·16-s + 0.229·17-s + 0.930·19-s + 2.35·20-s − 2.36·22-s − 0.0569·23-s + 0.345·25-s + 0.482·26-s + 0.456·29-s − 0.418·31-s − 0.0953·32-s − 0.399·34-s + 0.292·37-s − 1.61·38-s − 2.07·40-s + 1.00·41-s − 1.59·43-s + 2.75·44-s + 0.0991·46-s + 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.212221864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.212221864\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 4.51T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.945T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 + 0.273T + 23T^{2} \) |
| 29 | \( 1 - 2.46T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 1.78T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 1.50T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 0.945T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.752160834525286832663573195578, −7.78601868693527263361404939496, −7.19670423749994907545950686853, −6.39312857937413224909551460148, −5.92633241491292663868405926347, −4.84477572420187324365843646499, −3.55014080526982427557507275629, −2.45398168771716383522827811699, −1.65556924776515740713041065762, −0.879395160376564536799404295835,
0.879395160376564536799404295835, 1.65556924776515740713041065762, 2.45398168771716383522827811699, 3.55014080526982427557507275629, 4.84477572420187324365843646499, 5.92633241491292663868405926347, 6.39312857937413224909551460148, 7.19670423749994907545950686853, 7.78601868693527263361404939496, 8.752160834525286832663573195578
