L(s) = 1 | + (1.5 − 0.866i)3-s + (2.60 + 4.51i)5-s + 5.77·7-s + (1.5 − 2.59i)9-s + 14.2·11-s + (1.04 + 0.605i)13-s + (7.82 + 4.51i)15-s + (−0.848 − 1.46i)17-s + (−0.755 + 18.9i)19-s + (8.66 − 5.00i)21-s + (1.02 − 1.77i)23-s + (−1.09 + 1.89i)25-s − 5.19i·27-s + (−15.6 − 9.02i)29-s − 4.61i·31-s + ⋯ |
L(s) = 1 | + (0.5 − 0.288i)3-s + (0.521 + 0.903i)5-s + 0.825·7-s + (0.166 − 0.288i)9-s + 1.29·11-s + (0.0806 + 0.0465i)13-s + (0.521 + 0.301i)15-s + (−0.0499 − 0.0864i)17-s + (−0.0397 + 0.999i)19-s + (0.412 − 0.238i)21-s + (0.0445 − 0.0772i)23-s + (−0.0436 + 0.0756i)25-s − 0.192i·27-s + (−0.538 − 0.311i)29-s − 0.148i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.021401460\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.021401460\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 19 | \( 1 + (0.755 - 18.9i)T \) |
good | 5 | \( 1 + (-2.60 - 4.51i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 5.77T + 49T^{2} \) |
| 11 | \( 1 - 14.2T + 121T^{2} \) |
| 13 | \( 1 + (-1.04 - 0.605i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (0.848 + 1.46i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-1.02 + 1.77i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (15.6 + 9.02i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 4.61iT - 961T^{2} \) |
| 37 | \( 1 + 20.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-7.14 + 4.12i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-37.2 - 64.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (0.118 - 0.205i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (38.2 + 22.0i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-23.8 + 13.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (33.0 - 57.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-89.0 - 51.4i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (88.2 - 50.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (25.0 + 43.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-134. + 77.8i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 9.99T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-103. - 59.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (0.945 - 0.546i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.853139909126200929810104834144, −9.159232201374987325831243381516, −8.212011144429207157887522443323, −7.43049709851197203829227667683, −6.52443507202213712282861377157, −5.85382093741847229355013200628, −4.45094488558569275528728277564, −3.49796917186519104105597825283, −2.30253143528226635870388585050, −1.35508306687755303782302583662,
1.10284278255416948959813303264, 2.06997257068353461957602547380, 3.55890901551104713486305238851, 4.58069746447362537098836573340, 5.23543780677362667986982800654, 6.38376073307935287754294669611, 7.41574179579207533897631259876, 8.419673434135046250970080192775, 9.080809945922919309708175900589, 9.493474420610894183491880213186