L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.55 − 0.765i)3-s − 1.00i·4-s + (−1.73 − 1.41i)5-s + (−1.63 + 0.557i)6-s + (1.07 + 1.07i)7-s + (−0.707 − 0.707i)8-s + (1.82 + 2.37i)9-s + (−2.22 + 0.226i)10-s + 5.88i·11-s + (−0.765 + 1.55i)12-s + (−2.66 + 2.66i)13-s + 1.52·14-s + (1.61 + 3.52i)15-s − 1.00·16-s + (4.04 − 4.04i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.897 − 0.441i)3-s − 0.500i·4-s + (−0.775 − 0.631i)5-s + (−0.669 + 0.227i)6-s + (0.407 + 0.407i)7-s + (−0.250 − 0.250i)8-s + (0.609 + 0.792i)9-s + (−0.703 + 0.0715i)10-s + 1.77i·11-s + (−0.220 + 0.448i)12-s + (−0.738 + 0.738i)13-s + 0.407·14-s + (0.416 + 0.909i)15-s − 0.250·16-s + (0.980 − 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08963 + 0.107932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08963 + 0.107932i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (1.55 + 0.765i)T \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + (-1.07 - 1.07i)T + 7iT^{2} \) |
| 11 | \( 1 - 5.88iT - 11T^{2} \) |
| 13 | \( 1 + (2.66 - 2.66i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.04 + 4.04i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.95iT - 19T^{2} \) |
| 23 | \( 1 + (1.71 + 1.71i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.71T + 29T^{2} \) |
| 37 | \( 1 + (-6.38 - 6.38i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.84iT - 41T^{2} \) |
| 43 | \( 1 + (0.535 - 0.535i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.30 + 1.30i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.86 + 1.86i)T + 53iT^{2} \) |
| 59 | \( 1 + 6.57T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + (-8.88 - 8.88i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.12iT - 71T^{2} \) |
| 73 | \( 1 + (5.41 - 5.41i)T - 73iT^{2} \) |
| 79 | \( 1 - 16.1iT - 79T^{2} \) |
| 83 | \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.16T + 89T^{2} \) |
| 97 | \( 1 + (9.95 + 9.95i)T + 97iT^{2} \) |
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
−9.983249212201379202319720086620, −9.712534243038081711084946251528, −8.234694090435849613843113519915, −7.44708281851357158553436731483, −6.69908120917536910413242461767, −5.39258229419206885952534372877, −4.80534414195640961960839922052, −4.14518546040856321600293061112, −2.38535996549217044603909163804, −1.28151552667804795187456384026,
0.55329933175534286114624269538, 3.08183890688976418552326718107, 3.81962634183370742027684688456, 4.80532212142606995487216530399, 5.77950657404553914451910930369, 6.36775223750250462810631855127, 7.54072403602117118564994489287, 7.991132350807493356370736901160, 9.164594845024034881993956074293, 10.48051428375605006090417704420