L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.581 + 1.63i)3-s − 1.00i·4-s + (1.36 − 1.76i)5-s + (0.742 + 1.56i)6-s + (0.102 + 0.102i)7-s + (−0.707 − 0.707i)8-s + (−2.32 − 1.89i)9-s + (−0.282 − 2.21i)10-s − 6.12i·11-s + (1.63 + 0.581i)12-s + (−0.754 + 0.754i)13-s + 0.145·14-s + (2.08 + 3.26i)15-s − 1.00·16-s + (2.33 − 2.33i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.335 + 0.941i)3-s − 0.500i·4-s + (0.612 − 0.790i)5-s + (0.302 + 0.638i)6-s + (0.0388 + 0.0388i)7-s + (−0.250 − 0.250i)8-s + (−0.774 − 0.632i)9-s + (−0.0893 − 0.701i)10-s − 1.84i·11-s + (0.470 + 0.167i)12-s + (−0.209 + 0.209i)13-s + 0.0388·14-s + (0.539 + 0.842i)15-s − 0.250·16-s + (0.565 − 0.565i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42786 - 1.01198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42786 - 1.01198i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.581 - 1.63i)T \) |
| 5 | \( 1 + (-1.36 + 1.76i)T \) |
| 19 | \( 1 + iT \) |
good | 7 | \( 1 + (-0.102 - 0.102i)T + 7iT^{2} \) |
| 11 | \( 1 + 6.12iT - 11T^{2} \) |
| 13 | \( 1 + (0.754 - 0.754i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.33 + 2.33i)T - 17iT^{2} \) |
| 23 | \( 1 + (-6.21 - 6.21i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + (3.66 + 3.66i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + (0.994 - 0.994i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.88 + 4.88i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.86 + 2.86i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + (-4.37 - 4.37i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.311iT - 71T^{2} \) |
| 73 | \( 1 + (-0.755 + 0.755i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.3iT - 79T^{2} \) |
| 83 | \( 1 + (-11.1 - 11.1i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 + (0.595 + 0.595i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71095019118114029936488688392, −9.667494866274425806950141458691, −9.150385279465683601903817503559, −8.251059538883220634074575409277, −6.53355941265798988474028891667, −5.44517119865888835670182111096, −5.18237407435322550416174506469, −3.83229470529222203912499388600, −2.89015298111108977465857108119, −0.936245169091678200732961637435,
1.85573165322041027577092360839, 2.92445914629581224653929655827, 4.57855447129265411315535922495, 5.55979568385537762606746395864, 6.51653741536174390204458773202, 7.13569671340677378842761389096, 7.79786169090768890493140248064, 9.059489282193841346763485652694, 10.22638496137516407613067672074, 10.88073645700074864663028111574