| L(s) = 1 | + (1.40 + 0.153i)2-s + (−0.995 + 0.0980i)3-s + (1.95 + 0.431i)4-s + (−0.589 + 1.10i)5-s + (−1.41 − 0.0147i)6-s + (−0.582 + 2.93i)7-s + (2.67 + 0.905i)8-s + (0.980 − 0.195i)9-s + (−0.997 + 1.45i)10-s + (0.231 − 0.189i)11-s + (−1.98 − 0.237i)12-s + (−0.103 + 0.0553i)13-s + (−1.26 + 4.03i)14-s + (0.478 − 1.15i)15-s + (3.62 + 1.68i)16-s + (1.89 + 4.57i)17-s + ⋯ |
| L(s) = 1 | + (0.994 + 0.108i)2-s + (−0.574 + 0.0565i)3-s + (0.976 + 0.215i)4-s + (−0.263 + 0.493i)5-s + (−0.577 − 0.00604i)6-s + (−0.220 + 1.10i)7-s + (0.947 + 0.320i)8-s + (0.326 − 0.0650i)9-s + (−0.315 + 0.461i)10-s + (0.0696 − 0.0571i)11-s + (−0.573 − 0.0685i)12-s + (−0.0287 + 0.0153i)13-s + (−0.339 + 1.07i)14-s + (0.123 − 0.298i)15-s + (0.907 + 0.420i)16-s + (0.459 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66584 + 1.08422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66584 + 1.08422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.153i)T \) |
| 3 | \( 1 + (0.995 - 0.0980i)T \) |
| good | 5 | \( 1 + (0.589 - 1.10i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.582 - 2.93i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.231 + 0.189i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.103 - 0.0553i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 4.57i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-1.01 - 0.308i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (7.01 + 4.69i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-6.73 + 8.20i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.45 - 2.45i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.39 + 7.88i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.86 + 5.78i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (0.901 + 0.0888i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (5.26 - 2.18i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (5.74 + 7.00i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (2.09 + 1.12i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.688 - 6.98i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.178 + 1.80i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (4.83 + 0.961i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.07 + 5.39i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (5.60 + 2.31i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-4.35 + 14.3i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-4.55 + 3.04i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-11.4 - 11.4i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87075198372347241862079962536, −10.77751049389350062744398139398, −10.05229627511056812521694448425, −8.554852455073094840465845963802, −7.54888429474470004588530010910, −6.27370012006222195685644532092, −5.91851626391059253550358237559, −4.65561527971335576736303052764, −3.50074431748618256447131555531, −2.21086770193785912934426994351,
1.15575221002053767457489241331, 3.16614613120634546715457446731, 4.36059373059672648308422517272, 5.08241115260620665933977818532, 6.31266883339981897329605798434, 7.15790557151360520383597476744, 8.048608654812695997681503229520, 9.759909557919955232804276001244, 10.40554689087872536994561881568, 11.46932473305350487108729776448
