| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−7.28 + 5.29i)3-s + (−12.9 + 9.40i)4-s + (−49.3 + 26.1i)5-s + (29.1 + 21.1i)6-s − 91.4·7-s + (51.7 + 37.6i)8-s + (25.0 − 77.0i)9-s + (160. + 155. i)10-s + (173. + 534. i)11-s + (44.4 − 136. i)12-s + (−219. + 676. i)13-s + (112. + 347. i)14-s + (221. − 451. i)15-s + (79.1 − 243. i)16-s + (−1.79e3 − 1.30e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.883 + 0.468i)5-s + (0.330 + 0.239i)6-s − 0.705·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (0.508 + 0.491i)10-s + (0.433 + 1.33i)11-s + (0.0892 − 0.274i)12-s + (−0.360 + 1.11i)13-s + (0.154 + 0.474i)14-s + (0.253 − 0.518i)15-s + (0.0772 − 0.237i)16-s + (−1.50 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3555283954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3555283954\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 + (7.28 - 5.29i)T \) |
| 5 | \( 1 + (49.3 - 26.1i)T \) |
| good | 7 | \( 1 + 91.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-173. - 534. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (219. - 676. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.79e3 + 1.30e3i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-563. - 409. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-550. - 1.69e3i)T + (-5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-4.75e3 + 3.45e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (6.91e3 + 5.02e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-2.03e3 + 6.27e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.07e3 + 3.30e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.48e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.80e3 + 5.67e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (6.80e3 - 4.94e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-6.72e3 + 2.07e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.23e4 - 3.81e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (2.37e4 + 1.72e4i)T + (4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-5.69e4 + 4.13e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.00e4 + 3.10e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (3.28e4 - 2.38e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-7.02e4 - 5.10e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.00e4 - 9.25e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (4.74e4 - 3.45e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
| 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
−11.72737049219936683119050560378, −10.99194785309530847547589900942, −9.728003211164357529572977694008, −9.178543758086734828503836438235, −7.41699020308103227151881775945, −6.65436244256095786357796990447, −4.69093553928579855476868246597, −3.85294567146486387555077037107, −2.28586646843596619858463818565, −0.19469454966233030726771935874,
0.849472186288635844017104183861, 3.35730536825747997804490221767, 4.82894175770028323198798593869, 6.08385394767002337918284793664, 6.98527449719136885226497832864, 8.268657391209567374397172423707, 8.892789508588854166441092836967, 10.47251762206455491493592294104, 11.32690949670082363124623882364, 12.64125391205678634605097128054
