| L(s) = 1 | + (−0.445 − 1.94i)2-s + (−7.53 − 3.62i)3-s + (−3.60 + 1.73i)4-s + (2.19 + 9.59i)5-s + (−3.72 + 16.3i)6-s + (3.58 + 1.72i)7-s + (4.98 + 6.25i)8-s + (26.8 + 33.6i)9-s + (17.7 − 8.54i)10-s + (0.323 − 0.405i)11-s + 33.4·12-s + (−54.9 + 68.8i)13-s + (1.77 − 7.76i)14-s + (18.3 − 80.2i)15-s + (9.97 − 12.5i)16-s − 101.·17-s + ⋯ |
| L(s) = 1 | + (−0.157 − 0.689i)2-s + (−1.45 − 0.698i)3-s + (−0.450 + 0.216i)4-s + (0.195 + 0.858i)5-s + (−0.253 + 1.10i)6-s + (0.193 + 0.0933i)7-s + (0.220 + 0.276i)8-s + (0.992 + 1.24i)9-s + (0.560 − 0.270i)10-s + (0.00885 − 0.0111i)11-s + 0.805·12-s + (−1.17 + 1.46i)13-s + (0.0338 − 0.148i)14-s + (0.315 − 1.38i)15-s + (0.155 − 0.195i)16-s − 1.45·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.253271 + 0.216105i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.253271 + 0.216105i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.445 + 1.94i)T \) |
| 29 | \( 1 + (-45.3 - 149. i)T \) |
| good | 3 | \( 1 + (7.53 + 3.62i)T + (16.8 + 21.1i)T^{2} \) |
| 5 | \( 1 + (-2.19 - 9.59i)T + (-112. + 54.2i)T^{2} \) |
| 7 | \( 1 + (-3.58 - 1.72i)T + (213. + 268. i)T^{2} \) |
| 11 | \( 1 + (-0.323 + 0.405i)T + (-296. - 1.29e3i)T^{2} \) |
| 13 | \( 1 + (54.9 - 68.8i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + 101.T + 4.91e3T^{2} \) |
| 19 | \( 1 + (12.1 - 5.83i)T + (4.27e3 - 5.36e3i)T^{2} \) |
| 23 | \( 1 + (17.5 - 76.9i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 31 | \( 1 + (69.2 + 303. i)T + (-2.68e4 + 1.29e4i)T^{2} \) |
| 37 | \( 1 + (66.2 + 83.0i)T + (-1.12e4 + 4.93e4i)T^{2} \) |
| 41 | \( 1 - 47.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + (15.0 - 65.9i)T + (-7.16e4 - 3.44e4i)T^{2} \) |
| 47 | \( 1 + (302. - 379. i)T + (-2.31e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-123. - 541. i)T + (-1.34e5 + 6.45e4i)T^{2} \) |
| 59 | \( 1 + 269.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-484. - 233. i)T + (1.41e5 + 1.77e5i)T^{2} \) |
| 67 | \( 1 + (612. + 767. i)T + (-6.69e4 + 2.93e5i)T^{2} \) |
| 71 | \( 1 + (-4.52 + 5.67i)T + (-7.96e4 - 3.48e5i)T^{2} \) |
| 73 | \( 1 + (-79.0 + 346. i)T + (-3.50e5 - 1.68e5i)T^{2} \) |
| 79 | \( 1 + (-545. - 683. i)T + (-1.09e5 + 4.80e5i)T^{2} \) |
| 83 | \( 1 + (-804. + 387. i)T + (3.56e5 - 4.47e5i)T^{2} \) |
| 89 | \( 1 + (151. + 665. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + (-207. + 99.8i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−14.73137101743617049051407438181, −13.49860487665537604858011084647, −12.32005381820108833301404113781, −11.43559581541931473389459086331, −10.79148317285018862004285974327, −9.382437350487419003405366340537, −7.30752057783969630356189062909, −6.33833862339488567761725610896, −4.67988995702787311341231859629, −2.05982512430174440896169743274,
0.28805240011110930507424033909, 4.67558138384288204375360473904, 5.32920874927568617757330739726, 6.72796035087196738573379233537, 8.464627893919653287857171142707, 9.865368469872521181339427097437, 10.78355479607303013729093682137, 12.18415716084931593977987520325, 13.11937816594420558142464302972, 14.86250541113732487990371558005
