| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−2.59 − 1.5i)3-s + (−0.999 + 1.73i)4-s + (−1.48 + 4.77i)5-s + 4.24i·6-s + (−1.81 + 1.05i)7-s + 2.82·8-s + (4.5 + 7.79i)9-s + (6.89 − 1.55i)10-s + (−11.5 + 6.68i)11-s + (5.19 − 3.00i)12-s + (20.0 + 11.5i)13-s + (2.57 + 1.48i)14-s + (11.0 − 10.1i)15-s + (−2.00 − 3.46i)16-s − 27.5·17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.866 − 0.5i)3-s + (−0.249 + 0.433i)4-s + (−0.297 + 0.954i)5-s + 0.707i·6-s + (−0.259 + 0.150i)7-s + 0.353·8-s + (0.5 + 0.866i)9-s + (0.689 − 0.155i)10-s + (−1.05 + 0.607i)11-s + (0.433 − 0.250i)12-s + (1.54 + 0.890i)13-s + (0.183 + 0.106i)14-s + (0.735 − 0.677i)15-s + (−0.125 − 0.216i)16-s − 1.61·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.324594 + 0.285497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324594 + 0.285497i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 + (2.59 + 1.5i)T \) |
| 5 | \( 1 + (1.48 - 4.77i)T \) |
| good | 7 | \( 1 + (1.81 - 1.05i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (11.5 - 6.68i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-20.0 - 11.5i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 27.5T + 289T^{2} \) |
| 19 | \( 1 + 15.5T + 361T^{2} \) |
| 23 | \( 1 + (7.47 - 12.9i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-2.84 + 1.64i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-4.92 + 8.52i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 33.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-9.39 - 5.42i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-37.4 + 21.5i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (18.0 + 31.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 53.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-87.6 - 50.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (0.0505 + 0.0874i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (48.4 + 27.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 79.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 20.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-0.742 - 1.28i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (4.96 + 8.60i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-33.5 + 19.3i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.65208079382409353528295251029, −12.98134980085488048489248744019, −11.65391969971868436737754903310, −11.01890524228325483921387327730, −10.17493169068230900891544107810, −8.535567514586608839396356797968, −7.17556367487546698248201028482, −6.18336628790647960984669555171, −4.24052202179102709068732302169, −2.22040558796911556241639641067,
0.41787308777322131072163347470, 4.13116839415141454397131056197, 5.42872315185409107087711385738, 6.42909890583308634889271044001, 8.171526920668069783027975219317, 8.998045387260377605204920271019, 10.48889237908264551955019521317, 11.14474765716074733399202054938, 12.76852362224601810829881193633, 13.37789012351399090671358281571
