| L(s) = 1 | + (−0.756 + 0.549i)2-s + (2.59 + 7.97i)3-s + (−2.20 + 6.77i)4-s + (6.79 + 4.93i)5-s + (−6.33 − 4.60i)6-s + (2.16 − 6.65i)7-s + (−4.36 − 13.4i)8-s + (−35.0 + 25.4i)9-s − 7.85·10-s + (12.3 − 34.3i)11-s − 59.7·12-s + (0.809 − 0.587i)13-s + (2.02 + 6.22i)14-s + (−21.7 + 67.0i)15-s + (−35.4 − 25.7i)16-s + (74.7 + 54.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.267 + 0.194i)2-s + (0.498 + 1.53i)3-s + (−0.275 + 0.847i)4-s + (0.608 + 0.441i)5-s + (−0.431 − 0.313i)6-s + (0.116 − 0.359i)7-s + (−0.193 − 0.594i)8-s + (−1.29 + 0.942i)9-s − 0.248·10-s + (0.337 − 0.941i)11-s − 1.43·12-s + (0.0172 − 0.0125i)13-s + (0.0385 + 0.118i)14-s + (−0.374 + 1.15i)15-s + (−0.553 − 0.402i)16-s + (1.06 + 0.774i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.503568 + 1.41863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503568 + 1.41863i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.16 + 6.65i)T \) |
| 11 | \( 1 + (-12.3 + 34.3i)T \) |
| good | 2 | \( 1 + (0.756 - 0.549i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-2.59 - 7.97i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-6.79 - 4.93i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-74.7 - 54.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-19.6 - 60.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 76.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (31.6 - 97.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-141. + 102. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (32.1 - 98.8i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (111. + 344. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 290.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-119. - 368. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-120. + 87.8i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (255. - 787. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (298. + 216. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 554.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-683. - 496. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-288. + 887. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-1.04e3 + 758. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (274. + 199. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.19e3 + 869. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−14.32968863704620320563916455555, −13.78866038658776918480587090907, −12.15071934494027884952946690683, −10.66278082133276506184832377393, −9.891180133523513120147187502782, −8.849304012895782883842102090889, −7.84941952130344738491613972210, −5.93211949481389796705309407831, −4.14253837947385659619379182754, −3.19367770397125869408033949388,
1.11349652185017052000514527023, 2.27363791980936900304265630470, 5.20355388389625977249568330851, 6.46978960400563269934305704922, 7.77418368327942346563254609111, 9.054094676033686625863883631719, 9.867564174926773435150428737717, 11.64047872517154304383463970583, 12.55641073384693857676307774262, 13.65949772295660522439611385353
