| L(s) = 1 | + (1.39 − 0.221i)2-s + (0.786 − 1.54i)3-s + (1.90 − 0.618i)4-s + (4.42 + 2.33i)5-s + (0.756 − 2.32i)6-s + (0.284 − 0.284i)7-s + (2.52 − 1.28i)8-s + (−1.76 − 2.42i)9-s + (6.69 + 2.27i)10-s + (0.730 + 0.530i)11-s + (0.541 − 3.42i)12-s + (−3.27 − 0.519i)13-s + (0.334 − 0.460i)14-s + (7.07 − 4.99i)15-s + (3.23 − 2.35i)16-s + (0.365 + 0.716i)17-s + ⋯ |
| L(s) = 1 | + (0.698 − 0.110i)2-s + (0.262 − 0.514i)3-s + (0.475 − 0.154i)4-s + (0.884 + 0.466i)5-s + (0.126 − 0.388i)6-s + (0.0406 − 0.0406i)7-s + (0.315 − 0.160i)8-s + (−0.195 − 0.269i)9-s + (0.669 + 0.227i)10-s + (0.0664 + 0.0482i)11-s + (0.0451 − 0.285i)12-s + (−0.252 − 0.0399i)13-s + (0.0238 − 0.0328i)14-s + (0.471 − 0.332i)15-s + (0.202 − 0.146i)16-s + (0.0214 + 0.0421i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.47145 - 0.588131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47145 - 0.588131i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.221i)T \) |
| 3 | \( 1 + (-0.786 + 1.54i)T \) |
| 5 | \( 1 + (-4.42 - 2.33i)T \) |
| good | 7 | \( 1 + (-0.284 + 0.284i)T - 49iT^{2} \) |
| 11 | \( 1 + (-0.730 - 0.530i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.27 + 0.519i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.716i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (6.47 + 2.10i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (3.75 + 23.6i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (25.0 - 8.12i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (6.76 - 20.8i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (10.7 - 67.8i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (32.7 - 23.8i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (23.3 + 23.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.44 + 4.30i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-40.8 + 80.1i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (21.6 + 29.8i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (20.5 + 14.9i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-41.8 - 82.2i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (21.3 + 65.7i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (0.994 + 6.28i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (5.23 - 1.69i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-129. + 66.0i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-4.60 + 6.34i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-117. - 59.8i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−12.92251062253090105719738655166, −11.91141986913063928336433385723, −10.75489311604117081894761226718, −9.813303411325247381253753955761, −8.479849465587048151492817413647, −7.04929566554734790181011463627, −6.25716378329308219571558005870, −4.96540348183208445475711217961, −3.20178583077538906680313278044, −1.88165449365245597432366608238,
2.12612937728592946893903509071, 3.80155816979543587199847865966, 5.11690527135683639854301162208, 6.00588764686857840724598003860, 7.49314220262479426510575670042, 8.864617741558921935102670796167, 9.768659981273553925788544288151, 10.84207252902126576680668094158, 12.01871449282761735345085549552, 13.07904638943341082106458256361
