| L(s) = 1 | + (−0.951 − 0.309i)2-s + (−1.48 + 1.65i)3-s + (0.809 + 0.587i)4-s + (2.52 + 1.45i)5-s + (1.92 − 1.11i)6-s + (−3.86 − 0.406i)7-s + (−0.587 − 0.809i)8-s + (−0.202 − 1.92i)9-s + (−1.94 − 2.16i)10-s + (−0.401 − 0.902i)11-s + (−2.17 + 0.461i)12-s + (1.82 − 3.11i)13-s + (3.55 + 1.58i)14-s + (−6.14 + 1.99i)15-s + (0.309 + 0.951i)16-s + (−4.29 − 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (−0.858 + 0.953i)3-s + (0.404 + 0.293i)4-s + (1.12 + 0.650i)5-s + (0.785 − 0.453i)6-s + (−1.46 − 0.153i)7-s + (−0.207 − 0.286i)8-s + (−0.0673 − 0.641i)9-s + (−0.615 − 0.683i)10-s + (−0.121 − 0.272i)11-s + (−0.627 + 0.133i)12-s + (0.505 − 0.862i)13-s + (0.949 + 0.422i)14-s + (−1.58 + 0.515i)15-s + (0.0772 + 0.237i)16-s + (−1.04 − 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 806 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.315316 - 0.245833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.315316 - 0.245833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 13 | \( 1 + (-1.82 + 3.11i)T \) |
| 31 | \( 1 + (-3.55 - 4.28i)T \) |
| good | 3 | \( 1 + (1.48 - 1.65i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (-2.52 - 1.45i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.86 + 0.406i)T + (6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (0.401 + 0.902i)T + (-7.36 + 8.17i)T^{2} \) |
| 17 | \( 1 + (4.29 + 1.91i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (1.61 + 7.60i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (1.66 - 1.20i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.19 - 3.68i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.503 + 0.290i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.00 + 7.20i)T + (4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-4.74 + 1.00i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (3.57 - 1.16i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.0352 + 0.335i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (4.54 + 4.09i)T + (6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 + (6.65 + 3.84i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.28 + 0.345i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (6.72 + 15.0i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (3.05 + 1.36i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-8.04 + 7.24i)T + (8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (1.97 - 2.71i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.49 - 8.93i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.19958560795479802432023642009, −9.404031334563595924597332639696, −8.926765358765052976012404607873, −7.29832821235673513765588253916, −6.38582695698533185684558746428, −5.94092861541477787631235713644, −4.75676312926985182226045312902, −3.35446406673027393200354118547, −2.48852610205698001542742447386, −0.27960244472062597786158464265,
1.32503143883850222249213711181, 2.30426600066548430189936317318, 4.17322966465114745348648179617, 5.86801271221747847797598790702, 6.09409796468640550069401607538, 6.66153902270735835603875630246, 7.81644881093431328774956352837, 8.896270810519601331221162168148, 9.597444734870012159808165394896, 10.16941477380356662201177525827
