L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.274 − 1.71i)3-s − 1.00i·4-s + (2.05 + 0.893i)5-s + (1.40 + 1.01i)6-s + (2.76 + 2.76i)7-s + (0.707 + 0.707i)8-s + (−2.84 + 0.940i)9-s + (−2.08 + 0.818i)10-s + 4.37i·11-s + (−1.71 + 0.274i)12-s + (−0.0858 + 0.0858i)13-s − 3.91·14-s + (0.963 − 3.75i)15-s − 1.00·16-s + (−4.90 + 4.90i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.158 − 0.987i)3-s − 0.500i·4-s + (0.916 + 0.399i)5-s + (0.573 + 0.414i)6-s + (1.04 + 1.04i)7-s + (0.250 + 0.250i)8-s + (−0.949 + 0.313i)9-s + (−0.658 + 0.258i)10-s + 1.32i·11-s + (−0.493 + 0.0793i)12-s + (−0.0238 + 0.0238i)13-s − 1.04·14-s + (0.248 − 0.968i)15-s − 0.250·16-s + (−1.18 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05793 + 0.639620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05793 + 0.639620i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.274 + 1.71i)T \) |
| 5 | \( 1 + (-2.05 - 0.893i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-2.76 - 2.76i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.37iT - 11T^{2} \) |
| 13 | \( 1 + (0.0858 - 0.0858i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.90 - 4.90i)T - 17iT^{2} \) |
| 23 | \( 1 + (4.04 + 4.04i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4.80T + 31T^{2} \) |
| 37 | \( 1 + (1.15 + 1.15i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.99iT - 41T^{2} \) |
| 43 | \( 1 + (2.22 - 2.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.37 + 9.37i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.61 + 1.61i)T + 53iT^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + (9.57 + 9.57i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (3.04 - 3.04i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.24iT - 79T^{2} \) |
| 83 | \( 1 + (9.27 + 9.27i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + (1.05 + 1.05i)T + 97iT^{2} \) |
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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
−10.78416293714511673978814206396, −10.01261617796786835347168581020, −8.774023456496664502237781111063, −8.355508121520360050851526681023, −7.21505915607804996609071383482, −6.44417854704474972133356411812, −5.69175375102091638929931207150, −4.70143007212103969479268513016, −2.23224570528786533834322633129, −1.85326460056093780107634239531,
0.869626704867355035412930934756, 2.54179826417870870111736455119, 3.93161056460595743986462824579, 4.83529971211167423517186584802, 5.76896300426050096987428418756, 7.08854407531942653614837004118, 8.391719697135739618589740440707, 8.902597930104034263480760443752, 9.844661280453393128020319870178, 10.57207085385125412537880924050