| L(s) = 1 | + (−0.0174 + 1.99i)2-s + (2.53 − 1.60i)3-s + (−3.99 − 0.0696i)4-s + (0.754 − 4.94i)5-s + (3.16 + 5.09i)6-s + (−2.00 + 2.00i)7-s + (0.208 − 7.99i)8-s + (3.83 − 8.14i)9-s + (9.87 + 1.59i)10-s + (−6.53 − 4.74i)11-s + (−10.2 + 6.24i)12-s + (−0.223 + 1.41i)13-s + (−3.96 − 4.03i)14-s + (−6.02 − 13.7i)15-s + (15.9 + 0.556i)16-s + (−10.9 − 21.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.00870 + 0.999i)2-s + (0.844 − 0.535i)3-s + (−0.999 − 0.0174i)4-s + (0.150 − 0.988i)5-s + (0.528 + 0.849i)6-s + (−0.286 + 0.286i)7-s + (0.0261 − 0.999i)8-s + (0.426 − 0.904i)9-s + (0.987 + 0.159i)10-s + (−0.593 − 0.431i)11-s + (−0.853 + 0.520i)12-s + (−0.0172 + 0.108i)13-s + (−0.283 − 0.288i)14-s + (−0.401 − 0.915i)15-s + (0.999 + 0.0348i)16-s + (−0.644 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50745 - 0.658169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50745 - 0.658169i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0174 - 1.99i)T \) |
| 3 | \( 1 + (-2.53 + 1.60i)T \) |
| 5 | \( 1 + (-0.754 + 4.94i)T \) |
| good | 7 | \( 1 + (2.00 - 2.00i)T - 49iT^{2} \) |
| 11 | \( 1 + (6.53 + 4.74i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (0.223 - 1.41i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (10.9 + 21.5i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 6.60i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-32.6 + 5.17i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (7.65 + 23.5i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-29.6 - 9.63i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (32.6 + 5.16i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-10.7 - 14.8i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-11.7 - 11.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (10.0 - 19.7i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-27.2 + 53.5i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-47.1 - 64.9i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-75.0 - 54.4i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-40.9 - 80.2i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (36.9 + 113. i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (82.1 - 13.0i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (17.1 + 52.6i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-53.8 - 105. i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (41.0 + 29.8i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (3.97 - 7.80i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.68842970888038712300462588942, −9.939028679339842015000220883931, −9.050512074893515391173428914654, −8.608387222945362799022608592781, −7.56719279887522365220847334829, −6.66084051395585650660784601608, −5.45112720290261843884985437662, −4.41385287358830296305444712140, −2.81252479703918832705529596743, −0.74633325618334781770275917069,
2.01708353260485587032939294733, 3.10875123991410993521661849440, 3.98474930472895752385901450164, 5.28512929110835306332396299104, 6.94951592705947145538483080245, 8.087696031771129680887127771550, 9.039666430259245748126738830268, 10.10645687979571561260200781247, 10.48023597186638118127932489114, 11.31724542180898993799356174274
