L(s) = 1 | + (2.21 − 1.60i)2-s + (−2.44 − 7.54i)3-s + (−0.165 + 0.509i)4-s + (−12.0 − 8.73i)5-s + (−17.5 − 12.7i)6-s + (−0.949 + 2.92i)7-s + (7.20 + 22.1i)8-s + (−29.0 + 21.0i)9-s − 40.5·10-s + 4.24·12-s + (4.33 − 3.14i)13-s + (2.59 + 7.98i)14-s + (−36.3 + 112. i)15-s + (48.0 + 34.9i)16-s + (−33.3 − 24.2i)17-s + (−30.2 + 93.1i)18-s + ⋯ |
L(s) = 1 | + (0.781 − 0.567i)2-s + (−0.471 − 1.45i)3-s + (−0.0207 + 0.0637i)4-s + (−1.07 − 0.781i)5-s + (−1.19 − 0.866i)6-s + (−0.0512 + 0.157i)7-s + (0.318 + 0.980i)8-s + (−1.07 + 0.780i)9-s − 1.28·10-s + 0.102·12-s + (0.0924 − 0.0672i)13-s + (0.0495 + 0.152i)14-s + (−0.626 + 1.92i)15-s + (0.751 + 0.545i)16-s + (−0.475 − 0.345i)17-s + (−0.396 + 1.21i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.267360 + 0.888885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.267360 + 0.888885i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-2.21 + 1.60i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (2.44 + 7.54i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (12.0 + 8.73i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (0.949 - 2.92i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-4.33 + 3.14i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (33.3 + 24.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (43.2 + 133. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-7.72 + 23.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (25.4 - 18.5i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-4.06 + 12.5i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (80.6 + 248. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 57.7T + 7.95e4T^{2} \) |
| 47 | \( 1 + (106. + 327. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-277. + 201. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-27.3 + 84.0i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-597. - 434. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 342.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-167. - 121. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-312. + 961. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-1.04e3 + 760. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-357. - 259. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.08e3 - 791. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.30116957650838859477012161908, −11.84695894578520411992414334458, −11.01763950904457889998006129080, −8.766025772076802900021179495385, −7.918432335699735245122228367493, −6.84716652214070212210899814469, −5.28709608817165638603720893083, −4.07274174705893810407213793645, −2.28299947740226295908498580768, −0.39687397648272848317572567463,
3.72954752496905690904297636744, 4.20106109423146683416046633176, 5.56662477615206137443350964032, 6.64826663378829889024170167051, 8.069288637078392250859094068501, 9.756425622961613071310729114341, 10.50271752968581474754600475448, 11.35111419572107516753639408910, 12.54261906907941650946070936029, 14.05738938774043198239155209597