| L(s) = 1 | + (−1.96 − 3.26i)2-s + (1.61 − 2.52i)3-s + (−4.93 + 9.30i)4-s + (0.764 + 7.03i)5-s + (−11.4 − 0.298i)6-s + (−7.86 + 2.64i)7-s + (24.8 − 1.34i)8-s + (−3.79 − 8.16i)9-s + (21.4 − 16.3i)10-s + (11.9 + 8.12i)11-s + (15.5 + 27.4i)12-s + (2.93 + 2.78i)13-s + (24.1 + 20.4i)14-s + (19.0 + 9.40i)15-s + (−29.6 − 43.6i)16-s + (1.83 − 5.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.982 − 1.63i)2-s + (0.537 − 0.843i)3-s + (−1.23 + 2.32i)4-s + (0.152 + 1.40i)5-s + (−1.90 − 0.0496i)6-s + (−1.12 + 0.378i)7-s + (3.10 − 0.168i)8-s + (−0.421 − 0.906i)9-s + (2.14 − 1.63i)10-s + (1.08 + 0.738i)11-s + (1.29 + 2.29i)12-s + (0.226 + 0.214i)13-s + (1.72 + 1.46i)14-s + (1.26 + 0.627i)15-s + (−1.85 − 2.72i)16-s + (0.107 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.482i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.802862 - 0.206343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802862 - 0.206343i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.61 + 2.52i)T \) |
| 59 | \( 1 + (54.5 - 22.5i)T \) |
| good | 2 | \( 1 + (1.96 + 3.26i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.764 - 7.03i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (7.86 - 2.64i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (-11.9 - 8.12i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (-2.93 - 2.78i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-1.83 + 5.43i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 22.9i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (-28.1 + 7.81i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (18.9 - 31.4i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (6.27 - 38.2i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (1.73 - 32.0i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (-24.6 - 6.84i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (22.2 + 32.8i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (4.55 - 41.9i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-45.2 + 59.5i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-30.4 + 18.3i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-0.462 - 8.53i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (1.53 - 14.1i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-5.70 + 6.71i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (-11.3 + 28.3i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (9.32 + 20.1i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-38.2 + 63.5i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (-36.4 - 42.9i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−12.26621373255991725446660111247, −11.39640183188640619215798855013, −10.31151940899572275040062178301, −9.495100438525922955789320254087, −8.756960732568333077854714061717, −7.30918715324320019832492665797, −6.60572734405180931504152926475, −3.54363138863314473678526750224, −2.93720728532414931160552670837, −1.61262023132773535436647397009,
0.69821641469672070369789868265, 4.03158921182617461551982341679, 5.26551094623537925946632380549, 6.24728788566206251244291866807, 7.57992496722583343493433445034, 8.750608088246840626852610753494, 9.192283655979619627625531205722, 9.762577753364064489375278869403, 11.09560765187403615950822036821, 13.19700198193870590448442868679
