L(s) = 1 | + (−0.5 − 0.363i)2-s + (0.309 − 0.951i)3-s + (−0.5 − 1.53i)4-s + (2.11 − 1.53i)5-s + (−0.5 + 0.363i)6-s + (−0.927 − 2.85i)7-s + (−0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s − 1.61·10-s − 1.61·12-s + (1.42 + 1.03i)13-s + (−0.572 + 1.76i)14-s + (−0.809 − 2.48i)15-s + (−1.49 + 1.08i)16-s + (1.30 − 0.951i)17-s + (0.190 + 0.587i)18-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.256i)2-s + (0.178 − 0.549i)3-s + (−0.250 − 0.769i)4-s + (0.947 − 0.688i)5-s + (−0.204 + 0.148i)6-s + (−0.350 − 1.07i)7-s + (−0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s − 0.511·10-s − 0.467·12-s + (0.395 + 0.287i)13-s + (−0.153 + 0.471i)14-s + (−0.208 − 0.642i)15-s + (−0.374 + 0.272i)16-s + (0.317 − 0.230i)17-s + (0.0450 + 0.138i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.412995 - 1.09748i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.412995 - 1.09748i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (-2.11 + 1.53i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.927 + 2.85i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 1.03i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.30 + 0.951i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 + (1.38 + 4.25i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.30 + 1.67i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.0729 - 0.224i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.69 + 11.3i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 + 1.53i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.78 - 5.65i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.19 - 9.82i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-6.35 + 4.61i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 9.56T + 67T^{2} \) |
| 71 | \( 1 + (-4.5 + 3.26i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1 - 3.07i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.66 - 5.56i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.572 + 0.416i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 0.527T + 89T^{2} \) |
| 97 | \( 1 + (-11.3 - 8.24i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.80064310481753265910553532732, −10.09072182437223029648846060275, −9.329921962513959354053154094587, −8.522039856768935507812202544591, −7.29120252749070145096650346640, −6.14463405279921514306950701578, −5.37469693186460567870880355434, −3.94664202589201957480362358997, −2.02375002024863075118287765577, −0.928664665859263702013630722984,
2.53850750597214293231704952973, 3.38454046814789050640293100230, 4.97070354680024222411578640611, 6.12542947214131653677647841175, 6.99193715476250176634363956813, 8.333120030424266702211878757052, 9.058789271319819548418828106978, 9.694917092260331964034640553916, 10.70122351442075975151750186283, 11.68738350475949827684276087777