| L(s) = 1 | + (−1.97 + 0.299i)2-s + (1.39 − 2.65i)3-s + (3.82 − 1.18i)4-s + (0.0178 − 4.99i)5-s + (−1.97 + 5.66i)6-s + (8.85 − 8.85i)7-s + (−7.19 + 3.49i)8-s + (−5.08 − 7.42i)9-s + (1.46 + 9.89i)10-s + (2.81 − 8.67i)11-s + (2.19 − 11.7i)12-s + (3.74 + 1.90i)13-s + (−14.8 + 20.1i)14-s + (−13.2 − 7.04i)15-s + (13.1 − 9.05i)16-s + (−3.99 + 25.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.988 + 0.149i)2-s + (0.466 − 0.884i)3-s + (0.955 − 0.296i)4-s + (0.00356 − 0.999i)5-s + (−0.328 + 0.944i)6-s + (1.26 − 1.26i)7-s + (−0.899 + 0.436i)8-s + (−0.565 − 0.824i)9-s + (0.146 + 0.989i)10-s + (0.256 − 0.788i)11-s + (0.183 − 0.983i)12-s + (0.287 + 0.146i)13-s + (−1.06 + 1.44i)14-s + (−0.882 − 0.469i)15-s + (0.824 − 0.566i)16-s + (−0.234 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.660006 - 1.20449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.660006 - 1.20449i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.97 - 0.299i)T \) |
| 3 | \( 1 + (-1.39 + 2.65i)T \) |
| 5 | \( 1 + (-0.0178 + 4.99i)T \) |
| good | 7 | \( 1 + (-8.85 + 8.85i)T - 49iT^{2} \) |
| 11 | \( 1 + (-2.81 + 8.67i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-3.74 - 1.90i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (3.99 - 25.1i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-10.7 + 7.80i)T + (111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-12.4 - 24.5i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (10.9 + 7.95i)T + (259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-28.1 - 38.7i)T + (-296. + 913. i)T^{2} \) |
| 37 | \( 1 + (13.4 - 26.4i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-32.0 + 10.4i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (27.6 + 27.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (5.97 + 37.6i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (9.52 + 60.1i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-12.2 + 3.96i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (16.0 - 49.2i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (9.88 - 62.4i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-52.8 - 38.4i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-13.8 - 27.1i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (44.3 + 32.1i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (4.15 - 26.2i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-23.9 + 73.6i)T + (-6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-21.2 - 134. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.27429619360376154763615600780, −10.19614745193173837479487422611, −8.860064207082668152371696910964, −8.369455177601023350400517575034, −7.63513655767055933356496979619, −6.65125334594729474353181165657, −5.35882522702289060299212774778, −3.69776876302589669804661567353, −1.63451604088330370541337655191, −0.954883646244779760732030812940,
2.14288478366724106536255540178, 2.96282189527462801719071467546, 4.67667878224021469612650292269, 5.97948183163455983923072661105, 7.39852334705862625069056766033, 8.164208453945947778506762334740, 9.187246874717621997794886176140, 9.764397322847182751169588416116, 10.95522448766939739214430299330, 11.36814242817133718860383909112
