| L(s) = 1 | + (3.23 + 2.35i)2-s + (2.78 + 8.55i)3-s + (4.94 + 15.2i)4-s + (36.5 − 42.3i)5-s + (−11.1 + 34.2i)6-s − 114.·7-s + (−19.7 + 60.8i)8-s + (−65.5 + 47.6i)9-s + (217. − 51.0i)10-s + (405. + 294. i)11-s + (−116. + 84.6i)12-s + (−337. + 245. i)13-s + (−369. − 268. i)14-s + (463. + 194. i)15-s + (−207. + 150. i)16-s + (−576. + 1.77e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.653 − 0.757i)5-s + (−0.126 + 0.388i)6-s − 0.881·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.688 − 0.161i)10-s + (1.01 + 0.734i)11-s + (−0.233 + 0.169i)12-s + (−0.554 + 0.402i)13-s + (−0.504 − 0.366i)14-s + (0.532 + 0.223i)15-s + (−0.202 + 0.146i)16-s + (−0.484 + 1.49i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.481645422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481645422\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 5 | \( 1 + (-36.5 + 42.3i)T \) |
| good | 7 | \( 1 + 114.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-405. - 294. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (337. - 245. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (576. - 1.77e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (527. - 1.62e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-3.29e3 - 2.39e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-221. - 682. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.06e3 + 3.29e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.10e3 + 801. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (5.75e3 - 4.18e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.40e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.14e3 + 3.53e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-2.77e3 - 8.53e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.28e4 + 2.38e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.62e4 - 1.90e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.05e4 - 3.24e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (8.59e3 + 2.64e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.08e4 - 2.24e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (9.12e3 + 2.80e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-3.16e4 + 9.73e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-4.96e4 - 3.60e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.44e4 + 1.36e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
| 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
−12.73400890139904319273356611847, −11.72268340585351248288769346863, −10.14959814129124145565619874715, −9.402189704102410516435007580534, −8.443559259712298918901165963459, −6.84962902758135438826107923891, −5.90278474946754971257468226826, −4.64314578640709463473829828680, −3.62745523852869850018332719200, −1.82754483543395860247733962046,
0.65700874774908396708238080697, 2.48736163646682583953015841776, 3.26212282305078665743664256002, 5.10428775295461667814867254482, 6.59075408626442858583975346572, 6.87295030505250289965328301980, 8.887257751324250420166205832575, 9.747607436035469256638786562599, 10.92938581289326705210248976912, 11.78635255630013638286596566241
