L(s) = 1 | + i·3-s + (−2.17 + 0.539i)5-s − i·7-s − 9-s − 3.26·11-s + 0.340i·13-s + (−0.539 − 2.17i)15-s − 5.75i·17-s + 6.49·19-s + 21-s − 8.49i·23-s + (4.41 − 2.34i)25-s − i·27-s + 2·29-s − 8.34·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.970 + 0.241i)5-s − 0.377i·7-s − 0.333·9-s − 0.983·11-s + 0.0943i·13-s + (−0.139 − 0.560i)15-s − 1.39i·17-s + 1.49·19-s + 0.218·21-s − 1.77i·23-s + (0.883 − 0.468i)25-s − 0.192i·27-s + 0.371·29-s − 1.49·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.611374 - 0.478059i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611374 - 0.478059i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.17 - 0.539i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 - 0.340iT - 13T^{2} \) |
| 17 | \( 1 + 5.75iT - 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 8.49iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8.34T + 31T^{2} \) |
| 37 | \( 1 - 6.15iT - 37T^{2} \) |
| 41 | \( 1 - 0.340T + 41T^{2} \) |
| 43 | \( 1 + 8.68iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 8.34iT - 53T^{2} \) |
| 59 | \( 1 + 6.83T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 + 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 1.50iT - 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 6.49iT - 97T^{2} \) |
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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
−10.16485401585064830005927136948, −9.217329664915764519943558449288, −8.278685866262678935807692354259, −7.48187030861113889547466905074, −6.78093503272641747618098681878, −5.30320136825486485088160033048, −4.67591968158364271176945699112, −3.54130857908522219542163562190, −2.69414588236829131715019694196, −0.39833095299998878180378607508,
1.41841607886830970650161220133, 2.94387350978937265984482804805, 3.89318873789888755691836913592, 5.24909029607056907244009885272, 5.86930420697728439054924775455, 7.35034657913300192377078678145, 7.64940445911153382883191531202, 8.540309579413192805849759208910, 9.401666875185231456711072234114, 10.51098681163951271153799285683