| L(s) = 1 | + (−0.466 + 2.94i)3-s + (4.43 − 2.31i)5-s + (−6.13 − 6.13i)7-s + (0.0930 + 0.0302i)9-s + (4.35 + 13.4i)11-s + (7.93 + 15.5i)13-s + (4.75 + 14.1i)15-s + (−3.91 − 24.7i)17-s + (6.17 − 8.49i)19-s + (20.9 − 15.2i)21-s + (33.2 + 16.9i)23-s + (14.2 − 20.5i)25-s + (−12.3 + 24.1i)27-s + (28.6 + 39.4i)29-s + (3.88 + 2.82i)31-s + ⋯ |
| L(s) = 1 | + (−0.155 + 0.982i)3-s + (0.886 − 0.462i)5-s + (−0.877 − 0.877i)7-s + (0.0103 + 0.00335i)9-s + (0.396 + 1.21i)11-s + (0.610 + 1.19i)13-s + (0.316 + 0.942i)15-s + (−0.230 − 1.45i)17-s + (0.324 − 0.447i)19-s + (0.998 − 0.725i)21-s + (1.44 + 0.737i)23-s + (0.571 − 0.820i)25-s + (−0.456 + 0.895i)27-s + (0.989 + 1.36i)29-s + (0.125 + 0.0910i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.518 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.60901 + 0.905519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60901 + 0.905519i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.43 + 2.31i)T \) |
| good | 3 | \( 1 + (0.466 - 2.94i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (6.13 + 6.13i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.35 - 13.4i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-7.93 - 15.5i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (3.91 + 24.7i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-6.17 + 8.49i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-33.2 - 16.9i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-28.6 - 39.4i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (-3.88 - 2.82i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (14.4 - 7.38i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (22.5 - 69.3i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (2.30 - 2.30i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-10.5 - 1.66i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-3.89 + 24.5i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (4.38 + 1.42i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-16.1 - 49.6i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (17.6 + 111. i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-12.7 + 9.24i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (33.9 + 17.3i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (29.4 + 40.5i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-125. + 19.8i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (49.1 - 15.9i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (5.71 + 0.905i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−10.98679029716254306999704102222, −10.08207638746982152007188746094, −9.444570728737888778391010628917, −9.056396548750752376561163320361, −7.08144142911487366789339661454, −6.68138605424560554284205436909, −4.99960082911575208419217915943, −4.54350137388503519107943105931, −3.20541630123279953032412894304, −1.37411850531826589231970277103,
0.969081610314203434809904333961, 2.45927029636869705386829152488, 3.51870020756155014317003956730, 5.69708517868754285300456980112, 6.10350241088814044056592129954, 6.84260526460622693720861143273, 8.231782693198774477372888679684, 8.941513180212464609980258284692, 10.13252608619233058395410709222, 10.81991527404292893323711926102
