L(s) = 1 | − 3.12·5-s + i·7-s − 0.397i·11-s + 6.55i·13-s − 2i·17-s + 0.527·19-s − 8.21·23-s + 4.79·25-s − 4.73·29-s + 7.55i·31-s − 3.12i·35-s − 3.35i·37-s + 7.48i·41-s − 1.62·43-s + 10.4·47-s + ⋯ |
L(s) = 1 | − 1.39·5-s + 0.377i·7-s − 0.119i·11-s + 1.81i·13-s − 0.485i·17-s + 0.121·19-s − 1.71·23-s + 0.959·25-s − 0.878·29-s + 1.35i·31-s − 0.529i·35-s − 0.551i·37-s + 1.16i·41-s − 0.247·43-s + 1.52·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2104271540\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2104271540\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 3.12T + 5T^{2} \) |
| 11 | \( 1 + 0.397iT - 11T^{2} \) |
| 13 | \( 1 - 6.55iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 0.527T + 19T^{2} \) |
| 23 | \( 1 + 8.21T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 - 7.55iT - 31T^{2} \) |
| 37 | \( 1 + 3.35iT - 37T^{2} \) |
| 41 | \( 1 - 7.48iT - 41T^{2} \) |
| 43 | \( 1 + 1.62T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.795T + 53T^{2} \) |
| 59 | \( 1 + 2.47iT - 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 + 5.75T + 73T^{2} \) |
| 79 | \( 1 - 10.2iT - 79T^{2} \) |
| 83 | \( 1 - 8.36iT - 83T^{2} \) |
| 89 | \( 1 + 5.31iT - 89T^{2} \) |
| 97 | \( 1 + 19.1T + 97T^{2} \) |
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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
−7.966768400669809225781007456700, −7.11705301424882876973250167473, −6.67011152688101763185890623778, −5.72599942906645405925048758316, −4.82422923724890571517359810933, −4.08154265504746537696235383040, −3.65403790606603480354460767682, −2.52949190664231381423286323178, −1.56689526118957483960824107875, −0.07379981985015398068590179461,
0.78159452040336649814012626570, 2.20273953877551607169556194303, 3.26480054458226625485971338309, 3.90113513691384891723321538117, 4.40054558437863803066957637238, 5.56735987068300779325822747097, 5.97743426990485286342196464790, 7.22330003624073613499993957034, 7.59330328274373645636380135227, 8.137139049219144688216330436021