L(s) = 1 | + (0.178 + 1.72i)3-s + (−0.5 − 0.866i)5-s + (−0.178 + 0.308i)7-s + (−2.93 + 0.614i)9-s + (1.22 − 2.12i)11-s + (1.43 + 2.48i)13-s + (1.40 − 1.01i)15-s + 0.872·17-s + 7.34·19-s + (−0.563 − 0.252i)21-s + (−0.178 − 0.308i)23-s + (−0.499 + 0.866i)25-s + (−1.58 − 4.94i)27-s + (−4.37 + 7.57i)29-s + (4.38 + 7.59i)31-s + ⋯ |
L(s) = 1 | + (0.102 + 0.994i)3-s + (−0.223 − 0.387i)5-s + (−0.0673 + 0.116i)7-s + (−0.978 + 0.204i)9-s + (0.369 − 0.639i)11-s + (0.398 + 0.690i)13-s + (0.362 − 0.262i)15-s + 0.211·17-s + 1.68·19-s + (−0.122 − 0.0549i)21-s + (−0.0371 − 0.0643i)23-s + (−0.0999 + 0.173i)25-s + (−0.304 − 0.952i)27-s + (−0.812 + 1.40i)29-s + (0.787 + 1.36i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.550622577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.550622577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.178 - 1.72i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.178 - 0.308i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 2.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 2.48i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.872T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 + (0.178 + 0.308i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.38 - 7.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.87T + 37T^{2} \) |
| 41 | \( 1 + (-4.93 - 8.55i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.22 - 2.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.75 + 3.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + (-0.356 - 0.617i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.936 + 1.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.13 + 5.43i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + (6.32 - 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.50 + 11.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 1.25T + 89T^{2} \) |
| 97 | \( 1 + (6.87 - 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.562817188480488485937575713034, −9.025917665672280513139509238271, −8.402924542583494863496719035186, −7.40982219699540605517480042057, −6.33255553764108165107628371818, −5.39661328923122919513393356693, −4.73958276507077494831927007551, −3.64489062492511113299615876563, −3.05696734341137304727106070670, −1.29331881539908759541850336964,
0.69199618928708724499951044134, 2.02431205784507597977831943530, 3.09234699597166224377129512529, 4.01356569402469990347116658201, 5.44001472517646729606416325746, 6.06684659964481658241008783926, 7.14115748834264139773687198249, 7.54306346450424231929091647737, 8.306979062196634394664814920091, 9.332738611840774438835495404999