| L(s) = 1 | + (0.358 + 0.933i)2-s + (1.48 + 2.29i)3-s + (−0.743 + 0.669i)4-s + (−1.37 + 1.76i)5-s + (−1.60 + 2.20i)6-s + (1.94 + 1.79i)7-s + (−0.891 − 0.453i)8-s + (−1.81 + 4.07i)9-s + (−2.13 − 0.648i)10-s + (−3.05 + 1.30i)11-s + (−2.63 − 0.706i)12-s + (5.26 + 0.834i)13-s + (−0.976 + 2.45i)14-s + (−6.08 − 0.517i)15-s + (0.104 − 0.994i)16-s + (1.49 − 3.90i)17-s + ⋯ |
| L(s) = 1 | + (0.253 + 0.660i)2-s + (0.858 + 1.32i)3-s + (−0.371 + 0.334i)4-s + (−0.613 + 0.789i)5-s + (−0.655 + 0.902i)6-s + (0.735 + 0.677i)7-s + (−0.315 − 0.160i)8-s + (−0.604 + 1.35i)9-s + (−0.676 − 0.205i)10-s + (−0.919 + 0.392i)11-s + (−0.761 − 0.204i)12-s + (1.46 + 0.231i)13-s + (−0.261 + 0.657i)14-s + (−1.57 − 0.133i)15-s + (0.0261 − 0.248i)16-s + (0.363 − 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0215061 - 2.10382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0215061 - 2.10382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.358 - 0.933i)T \) |
| 5 | \( 1 + (1.37 - 1.76i)T \) |
| 7 | \( 1 + (-1.94 - 1.79i)T \) |
| 11 | \( 1 + (3.05 - 1.30i)T \) |
| good | 3 | \( 1 + (-1.48 - 2.29i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (-5.26 - 0.834i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.49 + 3.90i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-3.69 + 4.10i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (7.11 + 1.90i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-8.95 - 2.90i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.84 + 0.299i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (1.97 + 1.28i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (9.22 - 2.99i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.90 + 1.90i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.0548 - 1.04i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (0.414 + 0.335i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-1.94 - 2.15i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (13.4 + 1.41i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-6.41 + 1.71i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-4.88 - 3.54i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.536 - 10.2i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-5.51 + 12.3i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.34 + 14.8i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.275 - 0.476i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.695 - 4.39i)T + (-92.2 - 29.9i)T^{2} \) |
| 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
−10.56353123401526205420397279453, −9.871420295543622336388962577962, −8.793211618437573462832671943303, −8.295749786798006943874650971860, −7.52798657567354496769937170089, −6.34128410747513463962015095915, −5.10856750981074649050191894174, −4.48228043413952898874644749790, −3.39111576697621121662622514263, −2.63807994615218904539652633144,
0.988955881510562149293950069076, 1.73767856668467966381127191316, 3.26780349557585751533534414740, 4.02405770821887298250945376447, 5.35682365043896034791031301837, 6.38451834448303363863044518761, 7.87158644885835987267512622629, 8.062300512921148917798692030331, 8.596496249176529729293998780086, 10.02735089956925938581761248147
