L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.72 − 0.158i)3-s + (−0.499 + 0.866i)4-s + (−0.724 − 1.57i)6-s + (0.224 + 0.389i)7-s − 0.999·8-s + (2.94 + 0.548i)9-s + (1.72 + 2.98i)11-s + (0.999 − 1.41i)12-s + (−1.22 + 2.12i)13-s + (−0.224 + 0.389i)14-s + (−0.5 − 0.866i)16-s − 5.89·17-s + (0.999 + 2.82i)18-s − 5.44·19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.995 − 0.0917i)3-s + (−0.249 + 0.433i)4-s + (−0.295 − 0.642i)6-s + (0.0849 + 0.147i)7-s − 0.353·8-s + (0.983 + 0.182i)9-s + (0.520 + 0.900i)11-s + (0.288 − 0.408i)12-s + (−0.339 + 0.588i)13-s + (−0.0600 + 0.104i)14-s + (−0.125 − 0.216i)16-s − 1.43·17-s + (0.235 + 0.666i)18-s − 1.25·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.217122 + 0.795610i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.217122 + 0.795610i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.72 + 0.158i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.224 - 0.389i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.22 - 2.12i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.89T + 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + (3.44 - 5.97i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.775 - 1.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.22 - 3.85i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 + (6.62 - 11.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.27 - 3.94i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + (3.67 + 6.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 - 3.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + (6.5 + 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.61535572494109604270298756349, −10.71304789769732994936953785695, −9.638086324983497438561158772659, −8.739111279751943375408565598534, −7.40384526576942493299076264572, −6.73753042409328170542425827272, −5.93859080442780097752189167097, −4.73249618368452611864812295828, −4.13327770580837423371503938398, −1.98725334055698346864776450894,
0.50842669699888039317717842944, 2.33978666850735247412881546720, 4.03039882103822475083830124876, 4.70347292705058828804075686382, 6.06282639687909995976393198004, 6.51180642990837421445938485240, 8.051264339622026327627091388602, 9.116731229575190168853183349241, 10.20617403059933842579364491467, 10.87590542212028572586345440043