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
−15.85260845874832216201291189155, −14.80395367765084848088116569935, −13.61125533731548818343515555381, −12.77982084131360044009417679476, −11.13165208443043039062678621362, −9.454768517499498743535875657801, −8.468985607281277290593730468897, −6.16271692677866093089773720631, −5.21118236109139042911201762805, −1.83653963426987260236137631995,
3.13410729031416747220649615240, 4.85075632742316730143632564392, 7.14566249398237995335834626232, 8.575738439258230534627506281387, 10.29990452358594940538775087303, 11.09332218227775437639427189497, 13.22995059942401453618238054285, 13.63838058274282662095092514258, 15.08452290435747457822310249648, 16.54350434566011780511709011895