| L(s) = 1 | + (−1.26 + 0.642i)2-s + (1.71 − 0.270i)3-s + (1.17 − 1.61i)4-s + (3.78 − 3.27i)5-s + (−1.98 + 1.43i)6-s + (−0.746 − 0.746i)7-s + (−0.442 + 2.79i)8-s + (2.85 − 0.927i)9-s + (−2.66 + 6.54i)10-s + (3.90 − 12.0i)11-s + (1.57 − 3.08i)12-s + (−16.7 − 8.54i)13-s + (1.42 + 0.461i)14-s + (5.58 − 6.61i)15-s + (−1.23 − 3.80i)16-s + (26.7 + 4.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.321i)2-s + (0.570 − 0.0903i)3-s + (0.293 − 0.404i)4-s + (0.756 − 0.654i)5-s + (−0.330 + 0.239i)6-s + (−0.106 − 0.106i)7-s + (−0.0553 + 0.349i)8-s + (0.317 − 0.103i)9-s + (−0.266 + 0.654i)10-s + (0.355 − 1.09i)11-s + (0.131 − 0.257i)12-s + (−1.28 − 0.657i)13-s + (0.101 + 0.0329i)14-s + (0.372 − 0.441i)15-s + (−0.0772 − 0.237i)16-s + (1.57 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.38220 - 0.369074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38220 - 0.369074i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.642i)T \) |
| 3 | \( 1 + (-1.71 + 0.270i)T \) |
| 5 | \( 1 + (-3.78 + 3.27i)T \) |
| good | 7 | \( 1 + (0.746 + 0.746i)T + 49iT^{2} \) |
| 11 | \( 1 + (-3.90 + 12.0i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (16.7 + 8.54i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-26.7 - 4.24i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-12.4 - 17.0i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-3.67 - 7.20i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-19.1 + 26.2i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-1.25 + 0.911i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (27.5 - 54.0i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.80 - 5.57i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (45.2 - 45.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-5.31 - 33.5i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (89.9 - 14.2i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (9.18 - 2.98i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (7.59 - 23.3i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-45.8 - 7.26i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-16.1 - 11.7i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-12.1 - 23.7i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (46.8 - 64.4i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (10.3 - 65.4i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (36.6 + 11.9i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (18.7 + 118. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.74114557068700293460116350007, −11.77046273594714661620924207216, −10.02999976209811663906684018260, −9.776817298775865617181319529007, −8.428169723731215561565266434876, −7.75485163075280618491869209397, −6.22471603778404124211861175078, −5.17028088863356440418560419807, −3.10173408884247757751475251430, −1.22247838144110791902317102538,
1.94513477296490512295358275935, 3.15959325678278525263048319499, 5.02430400636494763376430461479, 6.86299189987748139871031716769, 7.48840546444331982776703124124, 9.131619554820162469330946872261, 9.709271940978704164208003649747, 10.48235066798145527028638450995, 11.89515986941998213317567593810, 12.67151106455651489673075827409
