| L(s) = 1 | + (0.723 + 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (3.28 − 2.58i)5-s + (0.841 + 0.540i)6-s + (2.43 + 1.04i)7-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (4.15 + 0.397i)10-s + (−1.89 + 1.80i)11-s + (0.235 + 0.971i)12-s + (2.09 + 4.57i)13-s + (1.03 + 2.43i)14-s + (2.73 − 3.15i)15-s + (−0.995 + 0.0950i)16-s + (−2.99 + 1.54i)17-s + ⋯ |
| L(s) = 1 | + (0.511 + 0.487i)2-s + (0.566 − 0.109i)3-s + (0.0237 + 0.499i)4-s + (1.46 − 1.15i)5-s + (0.343 + 0.220i)6-s + (0.918 + 0.394i)7-s + (−0.231 + 0.267i)8-s + (0.309 − 0.123i)9-s + (1.31 + 0.125i)10-s + (−0.569 + 0.543i)11-s + (0.0680 + 0.280i)12-s + (0.579 + 1.26i)13-s + (0.277 + 0.650i)14-s + (0.706 − 0.815i)15-s + (−0.248 + 0.0237i)16-s + (−0.727 + 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.25933 + 0.776967i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.25933 + 0.776967i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.723 - 0.690i)T \) |
| 3 | \( 1 + (-0.981 + 0.189i)T \) |
| 7 | \( 1 + (-2.43 - 1.04i)T \) |
| 23 | \( 1 + (4.70 - 0.944i)T \) |
| good | 5 | \( 1 + (-3.28 + 2.58i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (1.89 - 1.80i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.09 - 4.57i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.99 - 1.54i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (3.06 + 1.58i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (1.84 + 1.18i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.28 + 6.60i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (-11.2 + 4.49i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (-1.03 - 7.19i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.20 + 1.38i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (2.45 + 4.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.68 + 6.57i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (4.29 + 0.410i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (13.0 + 2.51i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (1.15 - 4.76i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (4.53 - 1.33i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.00355 - 0.0745i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (-5.48 + 7.70i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (-0.850 + 5.91i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (1.71 - 4.96i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-2.05 - 14.2i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−9.683617417997725983223002234709, −9.158785689253653009196619944772, −8.438756867105012958988203174209, −7.72080793494213586227251174281, −6.36807956000636685334884135155, −5.84314121964592128817449257790, −4.69845022855930030743607941573, −4.28834148511210779119093046891, −2.24047148765498162205720075287, −1.81984249333375997443761021276,
1.59168997849601616173940659056, 2.57026160970314389475622556918, 3.31313032611168286879854273147, 4.61069315885585998497352072438, 5.69066550304817614274039183272, 6.26669730128883788140174575288, 7.43665191845120757291430007086, 8.319329288058527536611404366302, 9.335109171714686851812905562670, 10.27606147946619818812833399547
