L(s) = 1 | + (0.5 − 0.866i)2-s + (−1.31 − 2.27i)3-s + (−0.499 − 0.866i)4-s + (−1.68 + 2.91i)5-s − 2.62·6-s + (−1.55 + 2.13i)7-s − 0.999·8-s + (−1.94 + 3.36i)9-s + (1.68 + 2.91i)10-s + (1.11 + 1.93i)11-s + (−1.31 + 2.27i)12-s − 3.10·13-s + (1.07 + 2.41i)14-s + 8.85·15-s + (−0.5 + 0.866i)16-s + (−3.18 − 5.51i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.757 − 1.31i)3-s + (−0.249 − 0.433i)4-s + (−0.753 + 1.30i)5-s − 1.07·6-s + (−0.589 + 0.807i)7-s − 0.353·8-s + (−0.648 + 1.12i)9-s + (0.533 + 0.923i)10-s + (0.335 + 0.581i)11-s + (−0.378 + 0.656i)12-s − 0.861·13-s + (0.286 + 0.646i)14-s + 2.28·15-s + (−0.125 + 0.216i)16-s + (−0.772 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0432 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0432 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.152397 + 0.145942i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.152397 + 0.145942i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.55 - 2.13i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.31 + 2.27i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.68 - 2.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.11 - 1.93i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 + (3.18 + 5.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.457 - 0.791i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 + (-1.73 - 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.70 - 8.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.85T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + (3.28 - 5.68i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.57 + 9.65i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.48 - 2.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.40 - 5.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.43 + 5.95i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.61T + 71T^{2} \) |
| 73 | \( 1 + (2.96 + 5.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.37 + 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.51T + 83T^{2} \) |
| 89 | \( 1 + (-0.792 + 1.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−11.81634398760448963222517867275, −11.41986847943831802504844903289, −10.29835934954092157671709187312, −9.212947934654729802875372558407, −7.61284806309302347412432522999, −6.90795716568955638792495282032, −6.22781501354075658829665801927, −4.89130851864802570996357816946, −3.17996774258039127593581343547, −2.15771577716793156206957020363,
0.14032784609561434340740717116, 3.84382252835713366516145792981, 4.24250361468833011411746974426, 5.22360038434488930780866150529, 6.22730840358083795177333246783, 7.54908699585792606486299850196, 8.697614925506222436550401153590, 9.435437183996506176516239417193, 10.52174528634585460325258661588, 11.35531656546841138949783136355