L(s) = 1 | + (0.523 − 0.380i)2-s + (0.927 + 2.85i)3-s + (−2.34 + 7.21i)4-s + (9.01 + 6.54i)5-s + (1.57 + 1.14i)6-s + (8.07 − 24.8i)7-s + (3.11 + 9.58i)8-s + (−7.28 + 5.29i)9-s + 7.20·10-s + (−36.0 + 5.31i)11-s − 22.7·12-s + (43.2 − 31.4i)13-s + (−5.22 − 16.0i)14-s + (−10.3 + 31.7i)15-s + (−43.7 − 31.8i)16-s + (−18.9 − 13.7i)17-s + ⋯ |
L(s) = 1 | + (0.185 − 0.134i)2-s + (0.178 + 0.549i)3-s + (−0.292 + 0.901i)4-s + (0.806 + 0.585i)5-s + (0.106 + 0.0776i)6-s + (0.436 − 1.34i)7-s + (0.137 + 0.423i)8-s + (−0.269 + 0.195i)9-s + 0.227·10-s + (−0.989 + 0.145i)11-s − 0.547·12-s + (0.923 − 0.671i)13-s + (−0.0997 − 0.307i)14-s + (−0.177 + 0.547i)15-s + (−0.684 − 0.497i)16-s + (−0.270 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.36194 + 0.536682i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36194 + 0.536682i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.927 - 2.85i)T \) |
| 11 | \( 1 + (36.0 - 5.31i)T \) |
good | 2 | \( 1 + (-0.523 + 0.380i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (-9.01 - 6.54i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-8.07 + 24.8i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-43.2 + 31.4i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (18.9 + 13.7i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (21.8 + 67.3i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (67.5 - 207. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-62.0 + 45.0i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (87.5 - 269. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-1.50 - 4.61i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 333.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (121. + 374. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (123. - 89.3i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (237. - 729. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-287. - 209. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (504. + 366. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (95.7 - 294. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-517. + 375. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (233. + 169. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 184.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (515. - 374. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−16.54350434566011780511709011895, −15.08452290435747457822310249648, −13.63838058274282662095092514258, −13.22995059942401453618238054285, −11.09332218227775437639427189497, −10.29990452358594940538775087303, −8.575738439258230534627506281387, −7.14566249398237995335834626232, −4.85075632742316730143632564392, −3.13410729031416747220649615240,
1.83653963426987260236137631995, 5.21118236109139042911201762805, 6.16271692677866093089773720631, 8.468985607281277290593730468897, 9.454768517499498743535875657801, 11.13165208443043039062678621362, 12.77982084131360044009417679476, 13.61125533731548818343515555381, 14.80395367765084848088116569935, 15.85260845874832216201291189155