| 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
−16.19077423767543189925522392922, −15.26668030668585810796653071417, −13.56672193231415608961893890487, −12.17035831802957753052766935559, −11.15527486697881906723455215948, −9.694940793618321799453086078041, −8.578821269398829469552882822942, −6.57237236415691067288561321909, −4.22036798168713370107747858254, −1.14424725613581804888388204731,
3.57983825342779251151980034020, 6.34760285294750310277789562108, 7.59639515224047130742223273565, 8.819313167363719462674985925172, 10.58671590118918502270512826215, 11.91118129406529661401077919323, 13.36106098523603079039005574431, 15.18399685781908963541346628297, 15.73688382830775430015064980316, 16.86607029007442859248972935677
