L(s) = 1 | + (−0.451 + 1.34i)2-s + (−2.93 + 0.624i)3-s + (−1.59 − 1.21i)4-s + (1.01 + 0.404i)5-s + (0.488 − 4.21i)6-s + (0.548 − 0.253i)7-s + (2.34 − 1.58i)8-s + (8.22 − 3.66i)9-s + (−1.00 + 1.17i)10-s + (9.41 + 2.61i)11-s + (5.42 + 2.55i)12-s + (−9.12 + 17.2i)13-s + (0.0924 + 0.849i)14-s + (−3.23 − 0.553i)15-s + (1.07 + 3.85i)16-s + (−3.82 + 8.26i)17-s + ⋯ |
L(s) = 1 | + (−0.225 + 0.670i)2-s + (−0.978 + 0.208i)3-s + (−0.398 − 0.302i)4-s + (0.203 + 0.0809i)5-s + (0.0813 − 0.702i)6-s + (0.0783 − 0.0362i)7-s + (0.292 − 0.198i)8-s + (0.913 − 0.407i)9-s + (−0.100 + 0.117i)10-s + (0.855 + 0.237i)11-s + (0.452 + 0.213i)12-s + (−0.701 + 1.32i)13-s + (0.00660 + 0.0606i)14-s + (−0.215 − 0.0369i)15-s + (0.0668 + 0.240i)16-s + (−0.225 + 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00658185 + 0.567570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00658185 + 0.567570i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (2.93 - 0.624i)T \) |
| 59 | \( 1 + (-11.0 - 57.9i)T \) |
good | 5 | \( 1 + (-1.01 - 0.404i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (-0.548 + 0.253i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-9.41 - 2.61i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (9.12 - 17.2i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (3.82 - 8.26i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-0.315 + 0.0693i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (3.16 + 0.519i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (3.01 + 8.95i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (16.6 + 3.66i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (25.4 - 37.4i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-2.53 + 0.415i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (-1.54 - 5.57i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (28.2 - 11.2i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (43.1 - 36.6i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (87.5 + 29.4i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-7.50 - 11.0i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-51.3 + 20.4i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (57.2 - 6.22i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-34.1 + 20.5i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-113. + 6.14i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-44.1 - 130. i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (88.5 + 9.62i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.72829494090966587539755857707, −10.73301013302331711387998900641, −9.725348848865572518471527516794, −9.152388730019923819970820319891, −7.74313194677370478719423559540, −6.67876977800115410498900719547, −6.17375760417906960117389494767, −4.86389161557337050772371628295, −4.06652842344908182177969283410, −1.64970919140898370256217280752,
0.32279040189197530520535530537, 1.80085499530220500186775404293, 3.48120361514817860563120777932, 4.88303302808349177747872236515, 5.73263459377932072992094479967, 6.98050233107532541421768455936, 7.940607850462972614674294278738, 9.234182199538365798249257296364, 10.02986949695107764653213366965, 10.91560130618299606366860074557