| L(s) = 1 | + (2.08 + 0.861i)2-s + (1.69 − 0.337i)3-s + (0.758 + 0.758i)4-s + (−2.21 + 3.30i)5-s + (3.82 + 0.761i)6-s + (−2.53 − 3.78i)7-s + (−2.52 − 6.09i)8-s + (2.77 − 1.14i)9-s + (−7.45 + 4.97i)10-s + (−1.75 + 8.82i)11-s + (1.54 + 1.03i)12-s + (−2.80 + 2.80i)13-s + (−2.00 − 10.0i)14-s + (−2.63 + 6.36i)15-s − 19.1i·16-s + (16.9 + 0.190i)17-s + ⋯ |
| L(s) = 1 | + (1.04 + 0.430i)2-s + (0.566 − 0.112i)3-s + (0.189 + 0.189i)4-s + (−0.442 + 0.661i)5-s + (0.637 + 0.126i)6-s + (−0.361 − 0.541i)7-s + (−0.315 − 0.761i)8-s + (0.307 − 0.127i)9-s + (−0.745 + 0.497i)10-s + (−0.159 + 0.801i)11-s + (0.128 + 0.0860i)12-s + (−0.215 + 0.215i)13-s + (−0.142 − 0.718i)14-s + (−0.175 + 0.424i)15-s − 1.19i·16-s + (0.999 + 0.0112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.79130 + 0.355275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79130 + 0.355275i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 + 0.337i)T \) |
| 17 | \( 1 + (-16.9 - 0.190i)T \) |
| good | 2 | \( 1 + (-2.08 - 0.861i)T + (2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (2.21 - 3.30i)T + (-9.56 - 23.0i)T^{2} \) |
| 7 | \( 1 + (2.53 + 3.78i)T + (-18.7 + 45.2i)T^{2} \) |
| 11 | \( 1 + (1.75 - 8.82i)T + (-111. - 46.3i)T^{2} \) |
| 13 | \( 1 + (2.80 - 2.80i)T - 169iT^{2} \) |
| 19 | \( 1 + (26.5 + 11.0i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-22.8 - 4.54i)T + (488. + 202. i)T^{2} \) |
| 29 | \( 1 + (-10.4 - 6.99i)T + (321. + 776. i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 6.11i)T + (-887. + 367. i)T^{2} \) |
| 37 | \( 1 + (24.7 - 4.91i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-26.3 - 39.4i)T + (-643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-75.5 + 31.2i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (31.3 - 31.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (83.3 + 34.5i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (28.1 + 68.0i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (46.4 - 31.0i)T + (1.42e3 - 3.43e3i)T^{2} \) |
| 67 | \( 1 + 87.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (23.1 - 4.61i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (49.1 - 73.6i)T + (-2.03e3 - 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-30.6 + 154. i)T + (-5.76e3 - 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-14.1 + 34.2i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-33.0 - 33.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-62.8 - 42.0i)T + (3.60e3 + 8.69e3i)T^{2} \) |
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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
−14.94844272223870369052880108628, −14.42894604765260142442624831960, −13.22390524328032900960379799072, −12.40295435688135285711437937035, −10.66951861256761860500560157089, −9.388061169454226939395501652178, −7.49414140625189076903878813636, −6.58044912776885680885475128170, −4.65659967978038810788727188818, −3.27317331012247819075371332847,
2.92289603274687745107901528544, 4.33922273669789205276566443629, 5.79871319907092862448003292348, 8.089709236388134096707835067924, 9.003161295870174766872468952398, 10.77694134375516379535602548706, 12.28683734686156294458674951656, 12.74145524646277364416474252604, 13.98124326004825238457433995918, 14.91005629688611644367424295507
