| L(s) = 1 | + (−0.533 + 2.68i)3-s + (−4.10 + 2.74i)5-s + (−5.75 − 3.84i)7-s + (1.39 + 0.579i)9-s + (−14.2 + 2.83i)11-s + (−8.40 − 8.40i)13-s + (−5.17 − 12.4i)15-s + (15.9 + 5.90i)17-s + (−19.3 + 8.03i)19-s + (13.3 − 13.3i)21-s + (4.46 + 22.4i)23-s + (−0.234 + 0.565i)25-s + (−15.9 + 23.9i)27-s + (−4.08 − 6.11i)29-s + (12.6 + 2.52i)31-s + ⋯ |
| L(s) = 1 | + (−0.177 + 0.894i)3-s + (−0.821 + 0.548i)5-s + (−0.822 − 0.549i)7-s + (0.155 + 0.0643i)9-s + (−1.29 + 0.257i)11-s + (−0.646 − 0.646i)13-s + (−0.344 − 0.832i)15-s + (0.937 + 0.347i)17-s + (−1.02 + 0.422i)19-s + (0.638 − 0.638i)21-s + (0.194 + 0.975i)23-s + (−0.00937 + 0.0226i)25-s + (−0.591 + 0.885i)27-s + (−0.140 − 0.210i)29-s + (0.408 + 0.0812i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 - 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0246907 + 0.484767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0246907 + 0.484767i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-15.9 - 5.90i)T \) |
| good | 3 | \( 1 + (0.533 - 2.68i)T + (-8.31 - 3.44i)T^{2} \) |
| 5 | \( 1 + (4.10 - 2.74i)T + (9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (5.75 + 3.84i)T + (18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (14.2 - 2.83i)T + (111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (8.40 + 8.40i)T + 169iT^{2} \) |
| 19 | \( 1 + (19.3 - 8.03i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-4.46 - 22.4i)T + (-488. + 202. i)T^{2} \) |
| 29 | \( 1 + (4.08 + 6.11i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-12.6 - 2.52i)T + (887. + 367. i)T^{2} \) |
| 37 | \( 1 + (2.00 - 10.0i)T + (-1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-60.2 - 40.2i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-27.8 - 11.5i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (40.3 + 40.3i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-54.7 + 22.6i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (2.81 - 6.79i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (45.0 - 67.4i)T + (-1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 58.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-2.20 + 11.0i)T + (-4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (60.7 - 40.6i)T + (2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (95.7 - 19.0i)T + (5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-31.0 - 74.8i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (117. - 117. i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (69.3 + 103. i)T + (-3.60e3 + 8.69e3i)T^{2} \) |
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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
−13.29112706339037866853526568361, −12.53369524020112989391305318179, −11.15126351126271420085495375668, −10.27093345072601995339700749287, −9.817169413443864819633743246220, −7.975223165900425018184711027179, −7.21374482205862976501414045741, −5.56570553872478844371086277170, −4.21487848926093992956988716771, −3.11191405634883168641304005162,
0.31713967176494509034077225076, 2.59364751356354760763259305911, 4.45688493625849787336962364830, 5.93990515467645326547048340215, 7.15135864266143245746150302535, 8.050618388918239513290190777756, 9.214888998781184318817854762108, 10.49092730897833894021630857249, 11.86996279667713188541220153932, 12.57829547052889788454595565452
