L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 5-s + (2.20 + 1.46i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s − 4.43i·11-s + (−4.51 + 2.60i)13-s + (2.64 + 0.172i)14-s + (−0.5 − 0.866i)16-s + (−3.46 − 6.00i)17-s + (−2.43 − 1.40i)19-s + (−0.499 + 0.866i)20-s + (−2.21 − 3.84i)22-s − 3.48i·23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.831 + 0.555i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s − 1.33i·11-s + (−1.25 + 0.723i)13-s + (0.705 + 0.0460i)14-s + (−0.125 − 0.216i)16-s + (−0.841 − 1.45i)17-s + (−0.557 − 0.321i)19-s + (−0.111 + 0.193i)20-s + (−0.473 − 0.819i)22-s − 0.726i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.710125236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.710125236\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.20 - 1.46i)T \) |
good | 11 | \( 1 + 4.43iT - 11T^{2} \) |
| 13 | \( 1 + (4.51 - 2.60i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.46 + 6.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.43 + 1.40i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.48iT - 23T^{2} \) |
| 29 | \( 1 + (-6.40 - 3.69i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.82 + 1.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.36 + 7.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.90 + 5.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.16 + 8.95i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.60 + 2.65i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.534 + 0.926i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.23 + 3.02i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.46 - 4.26i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (-13.1 + 7.61i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.88 + 6.72i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.84 + 4.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.16 - 7.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 6.62i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.854366352946343022063670125405, −8.334129262829280477897204395680, −7.18572662069576028142866734067, −6.60838200621224516324450025896, −5.41606360843120802953214056675, −4.85798880516213837862118528879, −4.07621742479441603486421276162, −2.82437611670937606162167685558, −2.17805492886181089697482226963, −0.48221508608464041761149572489,
1.63434542251137836136112571185, 2.70763314215815185389260270936, 4.06178591355026411011600026010, 4.52711312749187124154988125003, 5.25198310156635986105995866136, 6.44763550999012100116816660363, 7.09488945149232930748406785161, 8.046788345519887396885489876327, 8.160331755264361403436787438971, 9.608822199218139727003478428596