| L(s) = 1 | + (1.56 + 0.612i)2-s + (−3.71 − 3.63i)3-s + (−3.80 − 3.52i)4-s + (−5.95 − 4.05i)5-s + (−3.57 − 7.94i)6-s + (8.80 + 16.2i)7-s + (−9.59 − 19.9i)8-s + (0.643 + 26.9i)9-s + (−6.80 − 9.98i)10-s + (3.97 + 26.3i)11-s + (1.32 + 26.9i)12-s + (−18.0 + 14.3i)13-s + (3.76 + 30.8i)14-s + (7.40 + 36.7i)15-s + (0.332 + 4.44i)16-s + (23.7 + 7.31i)17-s + ⋯ |
| L(s) = 1 | + (0.551 + 0.216i)2-s + (−0.715 − 0.698i)3-s + (−0.475 − 0.441i)4-s + (−0.532 − 0.363i)5-s + (−0.243 − 0.540i)6-s + (0.475 + 0.879i)7-s + (−0.423 − 0.880i)8-s + (0.0238 + 0.999i)9-s + (−0.215 − 0.315i)10-s + (0.109 + 0.723i)11-s + (0.0319 + 0.647i)12-s + (−0.384 + 0.306i)13-s + (0.0719 + 0.588i)14-s + (0.127 + 0.631i)15-s + (0.00519 + 0.0693i)16-s + (0.338 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.209771 + 0.351933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.209771 + 0.351933i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.71 + 3.63i)T \) |
| 7 | \( 1 + (-8.80 - 16.2i)T \) |
| good | 2 | \( 1 + (-1.56 - 0.612i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (5.95 + 4.05i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (-3.97 - 26.3i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (18.0 - 14.3i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-23.7 - 7.31i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (113. - 65.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-41.5 - 134. i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-28.6 + 6.53i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (234. + 135. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (43.1 - 40.0i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (6.52 - 3.14i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (89.1 + 42.9i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-141. + 359. i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (485. - 523. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (178. - 121. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (133. + 144. i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (111. - 193. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-909. - 207. i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (344. - 135. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-18.5 - 32.0i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (49.7 - 62.4i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (418. + 63.1i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + 1.80e3iT - 9.12e5T^{2} \) |
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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
−12.61223569676517608656723761676, −12.31498964806258569706157684243, −11.20369007086996592273023323877, −9.882019789997338288918112315558, −8.640285189705589921474992752746, −7.47475767860116863136442338577, −6.15754877712913324209176601789, −5.25575149280549739456949027876, −4.24318392619758063571318933104, −1.75954995123239697763756838140,
0.18428008569184378661275421275, 3.27369234759306078413825350852, 4.26907737543432418789711409068, 5.17694830858928097126176538117, 6.72178463717633467801303676988, 8.066301657293681823928304795699, 9.188554819191057836494962693067, 10.73964729599931264376301525760, 11.08985028504057709679912201027, 12.24022718659049912918849316454
