L(s) = 1 | + (0.5 + 0.866i)2-s + (1.72 + 0.151i)3-s + (−0.499 + 0.866i)4-s + (−0.246 + 0.142i)5-s + (0.731 + 1.56i)6-s + (2.09 − 1.61i)7-s − 0.999·8-s + (2.95 + 0.522i)9-s + (−0.246 − 0.142i)10-s + (3.25 + 0.644i)11-s + (−0.993 + 1.41i)12-s − 6.02i·13-s + (2.44 + 1.00i)14-s + (−0.446 + 0.207i)15-s + (−0.5 − 0.866i)16-s + (−3.12 + 5.41i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.996 + 0.0873i)3-s + (−0.249 + 0.433i)4-s + (−0.110 + 0.0635i)5-s + (0.298 + 0.640i)6-s + (0.791 − 0.611i)7-s − 0.353·8-s + (0.984 + 0.174i)9-s + (−0.0778 − 0.0449i)10-s + (0.980 + 0.194i)11-s + (−0.286 + 0.409i)12-s − 1.67i·13-s + (0.654 + 0.268i)14-s + (−0.115 + 0.0536i)15-s + (−0.125 − 0.216i)16-s + (−0.758 + 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19405 + 0.972502i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19405 + 0.972502i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.72 - 0.151i)T \) |
| 7 | \( 1 + (-2.09 + 1.61i)T \) |
| 11 | \( 1 + (-3.25 - 0.644i)T \) |
good | 5 | \( 1 + (0.246 - 0.142i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 6.02iT - 13T^{2} \) |
| 17 | \( 1 + (3.12 - 5.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.70 - 2.71i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.55 + 0.898i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.434T + 29T^{2} \) |
| 31 | \( 1 + (3.64 - 6.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.326T + 41T^{2} \) |
| 43 | \( 1 + 2.29iT - 43T^{2} \) |
| 47 | \( 1 + (-3.52 + 2.03i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.66 - 0.960i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (8.38 + 4.83i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.37 + 1.95i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.34 + 7.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.88iT - 71T^{2} \) |
| 73 | \( 1 + (3.80 + 2.19i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.63 - 2.67i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 + (14.8 - 8.56i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−10.91140264340497272641498708266, −10.36287102255341944701512662547, −9.011344067827381057398496652159, −8.350635440059790603001000018948, −7.60302281078705482085894527317, −6.72830024307861201143990616084, −5.43308457766504081046534441721, −4.16513789159520299387175436167, −3.55517949172498706754913123820, −1.79835656025445780638507793517,
1.70788972864508989442160712049, 2.61290558149720372163521545574, 4.13761862510487074837311279821, 4.66328654118044228998795155107, 6.34320193265713035630283236335, 7.24831611039104433285935762901, 8.704779502018119308041430011829, 8.956070254860495546567572912832, 9.866052270262318724977308841902, 11.35278659080677425668586409215