| L(s) = 1 | + (−3.66 − 1.76i)2-s + (1.87 − 2.34i)3-s + (5.31 + 6.66i)4-s + (5.42 + 2.61i)5-s + (−10.9 + 5.29i)6-s + (−16.6 + 20.8i)7-s + (−0.477 − 2.09i)8-s + (−2.00 − 8.77i)9-s + (−15.2 − 19.1i)10-s + (2.60 − 11.3i)11-s + 25.5·12-s + (−19.2 + 84.5i)13-s + (97.9 − 47.1i)14-s + (16.2 − 7.83i)15-s + (13.2 − 58.0i)16-s + 83.2·17-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.623i)2-s + (0.359 − 0.451i)3-s + (0.664 + 0.833i)4-s + (0.485 + 0.233i)5-s + (−0.747 + 0.360i)6-s + (−0.899 + 1.12i)7-s + (−0.0211 − 0.0924i)8-s + (−0.0741 − 0.324i)9-s + (−0.482 − 0.605i)10-s + (0.0712 − 0.312i)11-s + 0.615·12-s + (−0.411 + 1.80i)13-s + (1.86 − 0.900i)14-s + (0.280 − 0.134i)15-s + (0.206 − 0.906i)16-s + 1.18·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.714001 + 0.200556i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.714001 + 0.200556i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.87 + 2.34i)T \) |
| 29 | \( 1 + (96.9 - 122. i)T \) |
| good | 2 | \( 1 + (3.66 + 1.76i)T + (4.98 + 6.25i)T^{2} \) |
| 5 | \( 1 + (-5.42 - 2.61i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 + (16.6 - 20.8i)T + (-76.3 - 334. i)T^{2} \) |
| 11 | \( 1 + (-2.60 + 11.3i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (19.2 - 84.5i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 - 83.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-75.0 - 94.1i)T + (-1.52e3 + 6.68e3i)T^{2} \) |
| 23 | \( 1 + (-72.4 + 34.8i)T + (7.58e3 - 9.51e3i)T^{2} \) |
| 31 | \( 1 + (29.0 + 13.9i)T + (1.85e4 + 2.32e4i)T^{2} \) |
| 37 | \( 1 + (-55.7 - 244. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-157. + 75.8i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-8.02 + 35.1i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-411. - 198. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + 218.T + 2.05e5T^{2} \) |
| 61 | \( 1 + (321. - 403. i)T + (-5.05e4 - 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-134. - 588. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (52.0 - 228. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-317. + 152. i)T + (2.42e5 - 3.04e5i)T^{2} \) |
| 79 | \( 1 + (187. + 820. i)T + (-4.44e5 + 2.13e5i)T^{2} \) |
| 83 | \( 1 + (521. + 654. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (329. + 158. i)T + (4.39e5 + 5.51e5i)T^{2} \) |
| 97 | \( 1 + (-173. - 217. i)T + (-2.03e5 + 8.89e5i)T^{2} \) |
| 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
−13.81044993219237301904476201134, −12.25876515191606886595605301811, −11.71966309571712397230513271027, −10.08504189370626500823243611048, −9.404183791693292561486962625881, −8.590055960993865088835626373234, −7.17405841851722769484033175360, −5.79976762905615130836598345808, −3.00637335406929655875083222628, −1.69424977175281343710933460745,
0.67626017841896644131533198988, 3.41224551420307675452950846515, 5.49717423552908046874367359174, 7.15750691304063249721210659348, 7.87171393259596281332867894968, 9.436257332424829080378485895368, 9.832718400026552036193590700441, 10.75761103885006424022715872116, 12.80902529059478312643298052363, 13.60386799917684741557293177751
