| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−0.977 − 1.42i)3-s + (0.939 − 0.342i)4-s + (0.640 + 2.14i)5-s + (1.21 + 1.23i)6-s + (1.22 + 0.709i)7-s + (−0.866 + 0.5i)8-s + (−1.08 + 2.79i)9-s + (−1.00 − 1.99i)10-s + (−4.15 + 2.40i)11-s + (−1.40 − 1.00i)12-s + (−4.38 − 3.68i)13-s + (−1.33 − 0.485i)14-s + (2.43 − 3.01i)15-s + (0.766 − 0.642i)16-s + (−0.869 − 4.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.564 − 0.825i)3-s + (0.469 − 0.171i)4-s + (0.286 + 0.958i)5-s + (0.494 + 0.505i)6-s + (0.464 + 0.268i)7-s + (−0.306 + 0.176i)8-s + (−0.363 + 0.931i)9-s + (−0.317 − 0.631i)10-s + (−1.25 + 0.723i)11-s + (−0.406 − 0.291i)12-s + (−1.21 − 1.02i)13-s + (−0.356 − 0.129i)14-s + (0.629 − 0.777i)15-s + (0.191 − 0.160i)16-s + (−0.210 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0103602 + 0.0871852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0103602 + 0.0871852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (0.977 + 1.42i)T \) |
| 5 | \( 1 + (-0.640 - 2.14i)T \) |
| 19 | \( 1 + (-3.92 - 1.90i)T \) |
| good | 7 | \( 1 + (-1.22 - 0.709i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.15 - 2.40i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (4.38 + 3.68i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.869 + 4.92i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (4.49 - 1.63i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.843 + 4.78i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (8.38 + 4.84i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.32T + 37T^{2} \) |
| 41 | \( 1 + (3.61 - 3.03i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.09 - 8.49i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.195 - 1.10i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.0601 + 0.165i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.697 - 3.95i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-0.668 + 0.243i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.07 - 11.7i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-11.1 - 4.05i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.496 - 0.591i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.74 - 8.03i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.06 - 1.84i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.424 + 0.356i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (3.40 + 19.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.17537990388581101718240140218, −10.08059644903521310323748180422, −9.793479360687760217149815499095, −8.039633381129559859712619316833, −7.58338856568396689876393697035, −6.97565454109885999760070285912, −5.67096602687506062228937665900, −5.13842183889946373791317325365, −2.80518485085665294257410159378, −2.03869588431879042617059055963,
0.06137072023992773344171664736, 1.88477923940145046778413595600, 3.61114508377540241702396808920, 4.91452255942665549452841132940, 5.43162718103781429638992852487, 6.73749670993823417558439173516, 7.919924681795432137202438800830, 8.786335292816782579496928985554, 9.445767622468587576328766131000, 10.43246765254828524687553821866
