| L(s) = 1 | + 32·2-s + 1.02e3·4-s + 4.58e3·5-s − 1.68e4·7-s + 3.27e4·8-s + 1.46e5·10-s + 5.22e5·11-s − 2.25e5·13-s − 5.37e5·14-s + 1.04e6·16-s + 7.10e6·17-s − 1.09e7·19-s + 4.68e6·20-s + 1.67e7·22-s + 9.53e6·23-s − 2.78e7·25-s − 7.22e6·26-s − 1.72e7·28-s + 4.12e6·29-s + 2.25e8·31-s + 3.35e7·32-s + 2.27e8·34-s − 7.69e7·35-s + 4.01e8·37-s − 3.48e8·38-s + 1.50e8·40-s + 4.49e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.655·5-s − 0.377·7-s + 0.353·8-s + 0.463·10-s + 0.978·11-s − 0.168·13-s − 0.267·14-s + 1/4·16-s + 1.21·17-s − 1.01·19-s + 0.327·20-s + 0.692·22-s + 0.308·23-s − 0.570·25-s − 0.119·26-s − 0.188·28-s + 0.0373·29-s + 1.41·31-s + 0.176·32-s + 0.858·34-s − 0.247·35-s + 0.951·37-s − 0.714·38-s + 0.231·40-s + 0.605·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.381342241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.381342241\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{5} T \) |
| good | 5 | \( 1 - 916 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 522754 T + p^{11} T^{2} \) |
| 13 | \( 1 + 225726 T + p^{11} T^{2} \) |
| 17 | \( 1 - 7106092 T + p^{11} T^{2} \) |
| 19 | \( 1 + 10905820 T + p^{11} T^{2} \) |
| 23 | \( 1 - 9537514 T + p^{11} T^{2} \) |
| 29 | \( 1 - 4122410 T + p^{11} T^{2} \) |
| 31 | \( 1 - 225621072 T + p^{11} T^{2} \) |
| 37 | \( 1 - 401516714 T + p^{11} T^{2} \) |
| 41 | \( 1 - 449235192 T + p^{11} T^{2} \) |
| 43 | \( 1 - 862042724 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2301032388 T + p^{11} T^{2} \) |
| 53 | \( 1 + 4033367010 T + p^{11} T^{2} \) |
| 59 | \( 1 - 4862200904 T + p^{11} T^{2} \) |
| 61 | \( 1 - 5473241454 T + p^{11} T^{2} \) |
| 67 | \( 1 + 5274256280 T + p^{11} T^{2} \) |
| 71 | \( 1 - 13847661862 T + p^{11} T^{2} \) |
| 73 | \( 1 + 12935724098 T + p^{11} T^{2} \) |
| 79 | \( 1 - 32414725668 T + p^{11} T^{2} \) |
| 83 | \( 1 - 30154096492 T + p^{11} T^{2} \) |
| 89 | \( 1 - 57844151544 T + p^{11} T^{2} \) |
| 97 | \( 1 - 70497350734 T + p^{11} T^{2} \) |
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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
−11.44233341748568037216269489029, −10.19161679592277513476248626195, −9.368380680697194601536244198363, −7.952940768602270024278575218523, −6.58276319574034873217686152855, −5.90582908780580560542246767921, −4.59224680990838754109742454684, −3.42689047225545304106315859834, −2.19960852501246044595339661334, −0.953613137943983098972503247991,
0.953613137943983098972503247991, 2.19960852501246044595339661334, 3.42689047225545304106315859834, 4.59224680990838754109742454684, 5.90582908780580560542246767921, 6.58276319574034873217686152855, 7.952940768602270024278575218523, 9.368380680697194601536244198363, 10.19161679592277513476248626195, 11.44233341748568037216269489029
