L(s) = 1 | + 2-s + 4-s + 5-s − 3.69i·7-s + 8-s + 10-s + 5.10·11-s − 6.05i·13-s − 3.69i·14-s + 16-s − 0.252i·17-s + 4.17·19-s + 20-s + 5.10·22-s + 5.65i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.39i·7-s + 0.353·8-s + 0.316·10-s + 1.53·11-s − 1.67i·13-s − 0.987i·14-s + 0.250·16-s − 0.0613i·17-s + 0.958·19-s + 0.223·20-s + 1.08·22-s + 1.17i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.913696089\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.913696089\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-6.06 + 5.50i)T \) |
good | 7 | \( 1 + 3.69iT - 7T^{2} \) |
| 11 | \( 1 - 5.10T + 11T^{2} \) |
| 13 | \( 1 + 6.05iT - 13T^{2} \) |
| 17 | \( 1 + 0.252iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + 9.12iT - 29T^{2} \) |
| 31 | \( 1 + 7.22iT - 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 43 | \( 1 + 5.80iT - 43T^{2} \) |
| 47 | \( 1 - 13.0iT - 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 2.77iT - 61T^{2} \) |
| 71 | \( 1 - 8.07iT - 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 8.22iT - 79T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 + 2.69iT - 89T^{2} \) |
| 97 | \( 1 + 3.99iT - 97T^{2} \) |
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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
−7.62431658280223559982534511960, −7.24573508441627629535885511443, −6.38392318955728755087478731366, −5.76270637627133625272336964298, −5.07327204808780911879145050482, −4.04241654194785772145129493494, −3.68377086082922754537573005537, −2.80649172456290090109153288779, −1.52354155169023185826703655570, −0.77352157494030169424186703797,
1.54996782822106979063967825805, 1.96521569315623276394850776645, 3.13448814056767649172365737759, 3.73562684884858951531251391561, 4.94398692397372693656176536484, 5.09974644448977863438507444362, 6.29320080051612579418314927032, 6.58485866330485018960350692716, 7.13657019235655530828016113551, 8.634366720700246567508335395509