| L(s) = 1 | + (−1.46 + 1.84i)2-s + (−0.0880 + 0.385i)3-s + (0.545 + 2.39i)4-s + (−7.25 + 9.10i)5-s + (−0.580 − 0.728i)6-s + (−1.72 + 7.57i)7-s + (−22.1 − 10.6i)8-s + (24.1 + 11.6i)9-s + (−6.10 − 26.7i)10-s + (20.0 − 9.65i)11-s − 0.970·12-s + (65.0 − 31.3i)13-s + (−11.4 − 14.3i)14-s + (−2.87 − 3.60i)15-s + (34.5 − 16.6i)16-s − 43.4·17-s + ⋯ |
| L(s) = 1 | + (−0.519 + 0.651i)2-s + (−0.0169 + 0.0742i)3-s + (0.0682 + 0.298i)4-s + (−0.649 + 0.814i)5-s + (−0.0395 − 0.0495i)6-s + (−0.0933 + 0.408i)7-s + (−0.980 − 0.472i)8-s + (0.895 + 0.431i)9-s + (−0.192 − 0.845i)10-s + (0.549 − 0.264i)11-s − 0.0233·12-s + (1.38 − 0.668i)13-s + (−0.217 − 0.272i)14-s + (−0.0494 − 0.0619i)15-s + (0.540 − 0.260i)16-s − 0.620·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.455047 + 0.737961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455047 + 0.737961i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (18.7 - 155. i)T \) |
| good | 2 | \( 1 + (1.46 - 1.84i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (0.0880 - 0.385i)T + (-24.3 - 11.7i)T^{2} \) |
| 5 | \( 1 + (7.25 - 9.10i)T + (-27.8 - 121. i)T^{2} \) |
| 7 | \( 1 + (1.72 - 7.57i)T + (-309. - 148. i)T^{2} \) |
| 11 | \( 1 + (-20.0 + 9.65i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-65.0 + 31.3i)T + (1.36e3 - 1.71e3i)T^{2} \) |
| 17 | \( 1 + 43.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-9.59 - 42.0i)T + (-6.17e3 + 2.97e3i)T^{2} \) |
| 23 | \( 1 + (12.2 + 15.3i)T + (-2.70e3 + 1.18e4i)T^{2} \) |
| 31 | \( 1 + (-126. + 158. i)T + (-6.62e3 - 2.90e4i)T^{2} \) |
| 37 | \( 1 + (-153. - 73.7i)T + (3.15e4 + 3.96e4i)T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (311. + 391. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-388. + 187. i)T + (6.47e4 - 8.11e4i)T^{2} \) |
| 53 | \( 1 + (128. - 161. i)T + (-3.31e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + 163.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (-5.56 + 24.4i)T + (-2.04e5 - 9.84e4i)T^{2} \) |
| 67 | \( 1 + (726. + 350. i)T + (1.87e5 + 2.35e5i)T^{2} \) |
| 71 | \( 1 + (-917. + 442. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (361. + 452. i)T + (-8.65e4 + 3.79e5i)T^{2} \) |
| 79 | \( 1 + (-91.2 - 43.9i)T + (3.07e5 + 3.85e5i)T^{2} \) |
| 83 | \( 1 + (239. + 1.04e3i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-928. + 1.16e3i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + (-159. - 699. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−16.86607029007442859248972935677, −15.73688382830775430015064980316, −15.18399685781908963541346628297, −13.36106098523603079039005574431, −11.91118129406529661401077919323, −10.58671590118918502270512826215, −8.819313167363719462674985925172, −7.59639515224047130742223273565, −6.34760285294750310277789562108, −3.57983825342779251151980034020,
1.14424725613581804888388204731, 4.22036798168713370107747858254, 6.57237236415691067288561321909, 8.578821269398829469552882822942, 9.694940793618321799453086078041, 11.15527486697881906723455215948, 12.17035831802957753052766935559, 13.56672193231415608961893890487, 15.26668030668585810796653071417, 16.19077423767543189925522392922
