| L(s) = 1 | + (3.23 + 2.35i)2-s + 24.2·3-s + (4.94 + 15.2i)4-s + (13.8 + 42.6i)5-s + (78.5 + 57.0i)6-s + (−139. + 101. i)7-s + (−19.7 + 60.8i)8-s + 345.·9-s + (−55.4 + 170. i)10-s + (−138. + 426. i)11-s + (119. + 369. i)12-s + (−170. − 123. i)13-s − 688.·14-s + (336. + 1.03e3i)15-s + (−207. + 150. i)16-s + (718. − 2.21e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + 1.55·3-s + (0.154 + 0.475i)4-s + (0.247 + 0.763i)5-s + (0.890 + 0.646i)6-s + (−1.07 + 0.780i)7-s + (−0.109 + 0.336i)8-s + 1.42·9-s + (−0.175 + 0.539i)10-s + (−0.345 + 1.06i)11-s + (0.240 + 0.740i)12-s + (−0.279 − 0.203i)13-s − 0.939·14-s + (0.385 + 1.18i)15-s + (−0.202 + 0.146i)16-s + (0.603 − 1.85i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0790 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0790 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.75397 + 2.54434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75397 + 2.54434i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 41 | \( 1 + (-9.80e3 - 4.43e3i)T \) |
| good | 3 | \( 1 - 24.2T + 243T^{2} \) |
| 5 | \( 1 + (-13.8 - 42.6i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (139. - 101. i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (138. - 426. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (170. + 123. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-718. + 2.21e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-2.26e3 + 1.64e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-2.76e3 - 2.00e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (2.01e3 + 6.20e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.65e3 - 5.08e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.35e3 - 7.25e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 43 | \( 1 + (-1.68e3 - 1.22e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (8.09e3 + 5.88e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (7.35e3 + 2.26e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.38e4 - 1.00e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (6.38e3 - 4.63e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.63e4 + 5.02e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (3.05e3 - 9.40e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + 3.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.88e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (9.86e3 - 7.16e3i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-3.19e4 - 9.84e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| 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
−13.70749444791753790698761387267, −13.01276377802168878098648442727, −11.74383397061675815560669155608, −9.716193914112772092131094285114, −9.316458252786022544736880855065, −7.60372358454929989971973823039, −6.91142419678718008503569789585, −5.11265372068342303222041524566, −3.04553089532548681693037826029, −2.72202531914366239615176121239,
1.25038734682127958959675298939, 3.07477169920116348377212327808, 3.87491800129964690795798433153, 5.76014698577234712325494083367, 7.47516413383373191022883556120, 8.731342231648490305397483744180, 9.655538736430132856274868108371, 10.71025959153566252327868388379, 12.72178975448371901084342637835, 13.02123357647765162630993789291
