L(s) = 1 | + (1.70 − 0.292i)3-s + (2 + 2i)7-s + (2.82 − i)9-s − 5.65i·11-s + (2.82 − 2.82i)17-s + 4i·19-s + (4 + 2.82i)21-s + (−4.24 − 4.24i)23-s + (4.53 − 2.53i)27-s + 5.65·29-s − 8·31-s + (−1.65 − 9.65i)33-s + (8 + 8i)37-s + 5.65i·41-s + (2 − 2i)43-s + ⋯ |
L(s) = 1 | + (0.985 − 0.169i)3-s + (0.755 + 0.755i)7-s + (0.942 − 0.333i)9-s − 1.70i·11-s + (0.685 − 0.685i)17-s + 0.917i·19-s + (0.872 + 0.617i)21-s + (−0.884 − 0.884i)23-s + (0.872 − 0.487i)27-s + 1.05·29-s − 1.43·31-s + (−0.288 − 1.68i)33-s + (1.31 + 1.31i)37-s + 0.883i·41-s + (0.304 − 0.304i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.611470456\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.611470456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.70 + 0.292i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (4.24 + 4.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-8 - 8i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-2 + 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.41 + 1.41i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.65 - 5.65i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-6 - 6i)T + 67iT^{2} \) |
| 71 | \( 1 - 11.3iT - 71T^{2} \) |
| 73 | \( 1 + (-8 + 8i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (9.89 + 9.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + (8 + 8i)T + 97iT^{2} \) |
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
−9.568351901633897184309750980201, −8.561281798706969262469938887517, −8.327362651389877490133045274105, −7.52324899056428097375967447055, −6.28702418207927132833153110643, −5.56177397386920940660470103374, −4.38935634780136278507682049183, −3.30567502074472137717414886927, −2.50276823803824224551266143531, −1.19623294082660006707349148348,
1.49355668061068021846717217508, 2.39960404982485733786715156638, 3.83289836781046076663451538097, 4.37721507507315872049635166110, 5.34461845825756923269238284504, 6.81067793268610892919957775478, 7.59230617220820557579783993920, 7.918857491858406637165221490827, 9.112804740348003121924971008834, 9.701252718173339607832131837081