| L(s) = 1 | − 1.49i·2-s − 0.248·4-s + 5-s + 0.522·7-s − 2.62i·8-s − 1.49i·10-s − 3.57i·11-s − 4.18·13-s − 0.783i·14-s − 4.43·16-s − 0.434i·17-s − 0.779i·19-s − 0.248·20-s − 5.36·22-s − 3.30·23-s + ⋯ |
| L(s) = 1 | − 1.06i·2-s − 0.124·4-s + 0.447·5-s + 0.197·7-s − 0.928i·8-s − 0.474i·10-s − 1.07i·11-s − 1.16·13-s − 0.209i·14-s − 1.10·16-s − 0.105i·17-s − 0.178i·19-s − 0.0554·20-s − 1.14·22-s − 0.689·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.649512002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.649512002\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + (-5.08 + 1.76i)T \) |
| good | 2 | \( 1 + 1.49iT - 2T^{2} \) |
| 7 | \( 1 - 0.522T + 7T^{2} \) |
| 11 | \( 1 + 3.57iT - 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 + 0.434iT - 17T^{2} \) |
| 19 | \( 1 + 0.779iT - 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 31 | \( 1 + 5.01iT - 31T^{2} \) |
| 37 | \( 1 - 0.232iT - 37T^{2} \) |
| 41 | \( 1 + 7.94iT - 41T^{2} \) |
| 43 | \( 1 - 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 4.66iT - 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.82T + 59T^{2} \) |
| 61 | \( 1 + 15.1iT - 61T^{2} \) |
| 67 | \( 1 + 2.28T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 4.99iT - 73T^{2} \) |
| 79 | \( 1 - 0.779iT - 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 2.19iT - 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−9.633996977526499359121710783078, −8.668496689656308321191435360981, −7.73175481991175576522012656912, −6.75658214416333152998428580892, −5.93596243126292232606363567312, −4.88858261053463674940912195788, −3.83135874195696419889986308203, −2.80180642836714844777338434005, −2.04858476779676954544920378846, −0.64516462390349420533659826536,
1.80968549024770878685810068534, 2.74596325215710833828610351377, 4.42086857142239374656218427468, 5.10923412014271393298163228054, 5.93973181580067828085064537219, 6.89720755625275663745376701187, 7.33294079130636297269242368819, 8.222932056105481942993118101626, 9.015083676632584701761231856793, 10.02371558413320648797469102820
