L(s) = 1 | + i·2-s + (0.866 + 0.5i)3-s − 4-s + (1.97 − 1.04i)5-s + (−0.5 + 0.866i)6-s + (−2.10 − 1.21i)7-s − i·8-s + (0.499 + 0.866i)9-s + (1.04 + 1.97i)10-s + (−0.345 − 0.598i)11-s + (−0.866 − 0.5i)12-s + (1.66 − 0.959i)13-s + (1.21 − 2.10i)14-s + (2.23 + 0.0874i)15-s + 16-s + (4.39 + 2.53i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.499 + 0.288i)3-s − 0.5·4-s + (0.884 − 0.465i)5-s + (−0.204 + 0.353i)6-s + (−0.796 − 0.460i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.329 + 0.625i)10-s + (−0.104 − 0.180i)11-s + (−0.249 − 0.144i)12-s + (0.460 − 0.266i)13-s + (0.325 − 0.563i)14-s + (0.576 + 0.0225i)15-s + 0.250·16-s + (1.06 + 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91774 + 0.632649i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91774 + 0.632649i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.97 + 1.04i)T \) |
| 31 | \( 1 + (-4.90 - 2.63i)T \) |
good | 7 | \( 1 + (2.10 + 1.21i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.345 + 0.598i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.66 + 0.959i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-4.39 - 2.53i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 3.71i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 6.84iT - 23T^{2} \) |
| 29 | \( 1 - 6.99T + 29T^{2} \) |
| 37 | \( 1 + (1.27 + 0.735i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.94 + 10.2i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.13 - 5.27i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.99iT - 47T^{2} \) |
| 53 | \( 1 + (-9.46 + 5.46i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.78 + 3.09i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + (12.2 - 7.09i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.66 + 9.80i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.34 - 3.08i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.37 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.756 + 0.436i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.31T + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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
−10.07199881328693871646010869974, −9.209967794024669190803050610372, −8.585217344407854335316717725868, −7.63804207178388357369447093945, −6.71796886239872257685949492184, −5.80865582995410356839998379302, −5.08450357301595354655836444917, −3.86000917894880882702432032257, −2.91815818605547732026305537769, −1.14499409644014240131132631921,
1.29190133518139796890750627879, 2.66058701003981838113631293706, 3.09880746387023927079744581924, 4.48714955281275439873821114116, 5.79550204648614551699864881125, 6.41438947238277636767381767599, 7.50891088974877702401977557509, 8.567797490280698123362803737594, 9.296817805455456666479331004512, 10.09269526328063764492630580156