| L(s) = 1 | + (0.978 − 2.36i)2-s + (0.0611 + 1.73i)3-s + (−3.20 − 3.20i)4-s + (−1.28 + 1.92i)5-s + (4.14 + 1.54i)6-s + (0.0883 − 0.0590i)7-s + (−5.98 + 2.47i)8-s + (−2.99 + 0.211i)9-s + (3.29 + 4.93i)10-s + (0.447 − 2.24i)11-s + (5.35 − 5.74i)12-s + (2.40 − 2.40i)13-s + (−0.0529 − 0.266i)14-s + (−3.41 − 2.11i)15-s + 7.49i·16-s + (2.22 + 3.47i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 1.66i)2-s + (0.0352 + 0.999i)3-s + (−1.60 − 1.60i)4-s + (−0.576 + 0.863i)5-s + (1.69 + 0.632i)6-s + (0.0333 − 0.0223i)7-s + (−2.11 + 0.876i)8-s + (−0.997 + 0.0705i)9-s + (1.04 + 1.55i)10-s + (0.134 − 0.677i)11-s + (1.54 − 1.65i)12-s + (0.666 − 0.666i)13-s + (−0.0141 − 0.0711i)14-s + (−0.882 − 0.545i)15-s + 1.87i·16-s + (0.539 + 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.809153 - 0.564691i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.809153 - 0.564691i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0611 - 1.73i)T \) |
| 17 | \( 1 + (-2.22 - 3.47i)T \) |
| good | 2 | \( 1 + (-0.978 + 2.36i)T + (-1.41 - 1.41i)T^{2} \) |
| 5 | \( 1 + (1.28 - 1.92i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.0883 + 0.0590i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.447 + 2.24i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.40 + 2.40i)T - 13iT^{2} \) |
| 19 | \( 1 + (1.76 + 0.729i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.88 + 0.773i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.33 - 2.22i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (4.29 - 0.854i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (0.207 + 1.04i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.15 - 6.21i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-5.47 + 2.26i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (2.07 + 2.07i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.47 + 10.7i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.49 + 3.51i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.123 + 0.185i)T + (-23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 - 6.91iT - 67T^{2} \) |
| 71 | \( 1 + (8.25 - 1.64i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.0499 - 0.0333i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (9.94 + 1.97i)T + (72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (7.97 + 3.30i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (7.83 - 7.83i)T - 89iT^{2} \) |
| 97 | \( 1 + (-7.02 + 10.5i)T + (-37.1 - 89.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.86050361553472414386784019184, −14.20702936626589198750130487323, −12.84669127430257141228079316123, −11.48717724104418747603406328572, −10.84360255991938014650403634896, −10.05609234829392936001587104187, −8.517132390087041149722729712677, −5.74298322682009967562289241838, −4.02356778791451945824150192242, −3.10692152472403573853374411726,
4.27233711152466692727587507438, 5.75801355122378882263420195288, 7.04363764622352039956386342333, 8.012479862512211809539866440536, 8.994516838788245639143326361062, 11.87947097409446641436004998602, 12.66551723985559887079270798800, 13.71484670766587122446140246895, 14.55097665308995126485831659247, 15.85146302352617850690222988397
