L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−2.15 + 0.587i)5-s + 1.83·7-s + (0.809 + 0.587i)8-s + (1.22 + 1.87i)10-s + (0.566 + 1.74i)11-s + (−0.747 + 2.29i)13-s + (−0.566 − 1.74i)14-s + (0.309 − 0.951i)16-s + (2.25 + 1.63i)17-s + (1.35 + 0.982i)19-s + (1.39 − 1.74i)20-s + (1.48 − 1.07i)22-s + (2.39 + 7.38i)23-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.964 + 0.262i)5-s + 0.692·7-s + (0.286 + 0.207i)8-s + (0.387 + 0.591i)10-s + (0.170 + 0.525i)11-s + (−0.207 + 0.637i)13-s + (−0.151 − 0.466i)14-s + (0.0772 − 0.237i)16-s + (0.546 + 0.396i)17-s + (0.310 + 0.225i)19-s + (0.313 − 0.389i)20-s + (0.316 − 0.229i)22-s + (0.500 + 1.54i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03026 + 0.145530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03026 + 0.145530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.15 - 0.587i)T \) |
good | 7 | \( 1 - 1.83T + 7T^{2} \) |
| 11 | \( 1 + (-0.566 - 1.74i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.747 - 2.29i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.25 - 1.63i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.35 - 0.982i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.39 - 7.38i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.13 + 4.45i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.28 - 3.11i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.406 - 1.25i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.08 - 3.34i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 + (-1.48 + 1.07i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.27 - 3.83i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.79 + 8.61i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.799 - 2.46i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (7.68 + 5.58i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.247 + 0.179i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 14.2i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.79 + 2.03i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.15 + 3.74i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.02 - 3.15i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.97 - 6.51i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.34851450721119146067624753459, −10.30115627165611347361536954611, −9.483854359177224141180986753319, −8.324841270069983281565088255613, −7.71850291502998222364306494019, −6.68806322238356209360309212887, −5.05981850346342886113287762348, −4.16667921516593358745001975001, −3.06099321466238628544769289602, −1.46639640858584809221119684048,
0.810720701918057698259400249482, 3.08747181847508739622640076525, 4.49860733272101886444310352838, 5.22395907222611187426636863847, 6.53850159441841130442668064804, 7.52484121723798891972586971819, 8.279657901952141419186403258659, 8.867472570729703288573169100646, 10.16694791823116845312379986082, 11.01992510490007913408179426106