L(s) = 1 | + (2.69 + 0.856i)2-s + (6.53 + 4.61i)4-s − 10.3i·5-s + (16.6 − 8.15i)7-s + (13.6 + 18.0i)8-s + (8.83 − 27.7i)10-s + 18.0i·11-s − 49.0i·13-s + (51.8 − 7.73i)14-s + (21.3 + 60.3i)16-s − 46.6i·17-s + 48.8·19-s + (47.6 − 67.3i)20-s + (−15.4 + 48.6i)22-s − 33.7i·23-s + ⋯ |
L(s) = 1 | + (0.952 + 0.302i)2-s + (0.816 + 0.577i)4-s − 0.921i·5-s + (0.897 − 0.440i)7-s + (0.603 + 0.797i)8-s + (0.279 − 0.878i)10-s + 0.494i·11-s − 1.04i·13-s + (0.989 − 0.147i)14-s + (0.332 + 0.942i)16-s − 0.666i·17-s + 0.589·19-s + (0.532 − 0.752i)20-s + (−0.149 + 0.471i)22-s − 0.306i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.641729728\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.641729728\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.69 - 0.856i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-16.6 + 8.15i)T \) |
good | 5 | \( 1 + 10.3iT - 125T^{2} \) |
| 11 | \( 1 - 18.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 49.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 48.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 409. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 470. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 203.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 493. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 240. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 995. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 790. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 885.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 67.3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 934. iT - 9.12e5T^{2} \) |
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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
−11.78153778990683086794247799199, −10.92488404970047201395287940642, −9.701210483293787480381149449957, −8.224481020121293104805500374189, −7.71389174202939848999540481922, −6.38548245088418542501121200136, −4.98193793653516233638517572549, −4.66296386125366916920216729536, −3.01347098163786011665957278543, −1.27368779829303292789361852688,
1.68340038714252014356930161032, 2.93687167606777282471949190126, 4.18730648640643140947112981291, 5.39907244343737742787511410691, 6.43199801585546278178060590653, 7.38166236311015842616349529427, 8.696199841731614847840138960700, 10.07415817881660272039606558188, 10.99636644688252927052219291748, 11.58917687456205967878603796587