| L(s) = 1 | + (−1.55 + 3.41i)2-s + (0.201 + 0.0592i)3-s + (−6.61 − 7.63i)4-s + (2.90 + 4.51i)5-s + (−0.517 + 0.597i)6-s + (7.13 + 1.02i)7-s + (21.9 − 6.45i)8-s + (−7.53 − 4.84i)9-s + (−19.9 + 2.86i)10-s + (5.58 − 2.55i)11-s + (−0.882 − 1.93i)12-s + (−1.22 − 8.51i)13-s + (−14.6 + 22.7i)14-s + (0.318 + 1.08i)15-s + (−6.48 + 45.1i)16-s + (−13.2 − 11.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.779 + 1.70i)2-s + (0.0673 + 0.0197i)3-s + (−1.65 − 1.90i)4-s + (0.580 + 0.903i)5-s + (−0.0862 + 0.0995i)6-s + (1.01 + 0.146i)7-s + (2.74 − 0.806i)8-s + (−0.837 − 0.537i)9-s + (−1.99 + 0.286i)10-s + (0.507 − 0.231i)11-s + (−0.0735 − 0.161i)12-s + (−0.0942 − 0.655i)13-s + (−1.04 + 1.62i)14-s + (0.0212 + 0.0722i)15-s + (−0.405 + 2.81i)16-s + (−0.777 − 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.347348 + 0.575071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.347348 + 0.575071i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-1.58 - 22.9i)T \) |
| good | 2 | \( 1 + (1.55 - 3.41i)T + (-2.61 - 3.02i)T^{2} \) |
| 3 | \( 1 + (-0.201 - 0.0592i)T + (7.57 + 4.86i)T^{2} \) |
| 5 | \( 1 + (-2.90 - 4.51i)T + (-10.3 + 22.7i)T^{2} \) |
| 7 | \( 1 + (-7.13 - 1.02i)T + (47.0 + 13.8i)T^{2} \) |
| 11 | \( 1 + (-5.58 + 2.55i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (1.22 + 8.51i)T + (-162. + 47.6i)T^{2} \) |
| 17 | \( 1 + (13.2 + 11.4i)T + (41.1 + 286. i)T^{2} \) |
| 19 | \( 1 + (2.09 - 1.81i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (4.81 - 5.55i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (8.51 - 2.49i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-11.8 + 18.4i)T + (-568. - 1.24e3i)T^{2} \) |
| 41 | \( 1 + (54.4 - 35.0i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-7.49 + 25.5i)T + (-1.55e3 - 9.99e2i)T^{2} \) |
| 47 | \( 1 - 49.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + (61.2 + 8.80i)T + (2.69e3 + 791. i)T^{2} \) |
| 59 | \( 1 + (1.09 + 7.59i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (7.78 + 26.5i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-81.5 - 37.2i)T + (2.93e3 + 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-23.9 + 52.4i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-79.6 - 91.9i)T + (-758. + 5.27e3i)T^{2} \) |
| 79 | \( 1 + (84.6 - 12.1i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (80.7 - 125. i)T + (-2.86e3 - 6.26e3i)T^{2} \) |
| 89 | \( 1 + (2.59 - 8.83i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-78.1 - 121. i)T + (-3.90e3 + 8.55e3i)T^{2} \) |
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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
−17.76665222796417355054879529053, −17.13348397748927551870544449436, −15.48598122822471068961812863053, −14.58579644851167868021525426998, −13.89798309920680849592464831497, −11.10141492615346912889485623479, −9.506296466776832790685104474204, −8.269926894913446585031462401194, −6.78771743189252291111341317751, −5.48697374580010449928025555585,
1.88027869582802873957658173957, 4.59979870376599271102837208914, 8.324178990205938562818381024194, 9.162534833970326074022635906167, 10.73280712951589913828144626395, 11.74440115439526897480999486311, 13.01463386555981916660944693987, 14.18988030397129780901805916125, 16.99285285116937342674325207861, 17.29507875590912565204027125665
