| L(s) = 1 | + (0.714 − 1.48i)2-s + (−7.48 − 5.96i)3-s + (18.2 + 22.8i)4-s + (37.6 + 18.1i)5-s + (−14.1 + 6.83i)6-s + (122. − 153. i)7-s + (98.3 − 22.4i)8-s + (−33.6 − 147. i)9-s + (53.8 − 42.9i)10-s + (353. + 80.6i)11-s − 280. i·12-s + (−205. + 898. i)13-s + (−140. − 292. i)14-s + (−173. − 360. i)15-s + (−171. + 751. i)16-s − 500. i·17-s + ⋯ |
| L(s) = 1 | + (0.126 − 0.262i)2-s + (−0.480 − 0.382i)3-s + (0.570 + 0.715i)4-s + (0.673 + 0.324i)5-s + (−0.161 + 0.0775i)6-s + (0.947 − 1.18i)7-s + (0.543 − 0.124i)8-s + (−0.138 − 0.607i)9-s + (0.170 − 0.135i)10-s + (0.880 + 0.200i)11-s − 0.561i·12-s + (−0.336 + 1.47i)13-s + (−0.191 − 0.398i)14-s + (−0.199 − 0.413i)15-s + (−0.167 + 0.734i)16-s − 0.420i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.80683 - 0.383782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80683 - 0.383782i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (3.64e3 + 2.68e3i)T \) |
| good | 2 | \( 1 + (-0.714 + 1.48i)T + (-19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (7.48 + 5.96i)T + (54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-37.6 - 18.1i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + (-122. + 153. i)T + (-3.73e3 - 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-353. - 80.6i)T + (1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (205. - 898. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + 500. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.20e3 + 962. i)T + (5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (3.82e3 - 1.84e3i)T + (4.01e6 - 5.03e6i)T^{2} \) |
| 31 | \( 1 + (-626. + 1.30e3i)T + (-1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (1.95e3 - 446. i)T + (6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 4.86e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-9.55e3 - 1.98e4i)T + (-9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (2.61e4 + 5.95e3i)T + (2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.92e3 - 1.40e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + 9.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.43e4 - 1.14e4i)T + (1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-8.29e3 - 3.63e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-746. + 3.27e3i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (1.00e4 + 2.07e4i)T + (-1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (-7.84e4 + 1.79e4i)T + (2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (5.18e4 + 6.49e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (1.25e4 - 2.60e4i)T + (-3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (3.16e4 - 2.52e4i)T + (1.91e9 - 8.37e9i)T^{2} \) |
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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
−16.36869122404698447587161122531, −14.43412526837798022951581635429, −13.57296795215093316573247052468, −11.83151069547593951398288364185, −11.37794234123437306703244542663, −9.631197776046733713577311803693, −7.50484163913383533843768127897, −6.50315911140875554378455611211, −4.09641889185505843884613562824, −1.68136654320527254284245446314,
1.87200491661796231105091431070, 5.25414160656332540209485554292, 5.84071430525828777059434206240, 8.102989080781442987192442426216, 9.839049799292847189386501979265, 11.01920453297567930639907888531, 12.19509634558920958640342345136, 14.04131533062644223240862937448, 14.99507550250731688903923244772, 16.08268492916350426373126483212
