L(s) = 1 | + (1.76 + 3.05i)2-s + (−2.20 + 3.82i)4-s + (−2.5 + 4.33i)5-s + (−12.7 − 22.0i)7-s + 12.6·8-s − 17.6·10-s + (35.6 + 61.7i)11-s + (25.6 − 44.5i)13-s + (44.8 − 77.6i)14-s + (39.9 + 69.1i)16-s − 33.3·17-s + 113.·19-s + (−11.0 − 19.1i)20-s + (−125. + 217. i)22-s + (−40.9 + 70.9i)23-s + ⋯ |
L(s) = 1 | + (0.622 + 1.07i)2-s + (−0.275 + 0.477i)4-s + (−0.223 + 0.387i)5-s + (−0.686 − 1.18i)7-s + 0.558·8-s − 0.557·10-s + (0.977 + 1.69i)11-s + (0.548 − 0.949i)13-s + (0.855 − 1.48i)14-s + (0.623 + 1.08i)16-s − 0.475·17-s + 1.36·19-s + (−0.123 − 0.213i)20-s + (−1.21 + 2.11i)22-s + (−0.371 + 0.643i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.763288948\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.763288948\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (-1.76 - 3.05i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (12.7 + 22.0i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-35.6 - 61.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-25.6 + 44.5i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + 33.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (40.9 - 70.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-123. - 213. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (111. - 192. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 22.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-217. + 376. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-118. - 205. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-53.9 - 93.5i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (85.5 - 148. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-39.7 - 68.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-305. + 529. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 511.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-396. - 687. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (135. + 233. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 177.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-440. - 763. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.90398021582376262127663176992, −10.23066914252154552967532681609, −9.290166646308382204843346170797, −7.75204321483428634351130375760, −7.10874456113919603846184183476, −6.62912069454886254751876933617, −5.37512536041597489741245992512, −4.27922892338312816076940371245, −3.43158062442353425444694886287, −1.29472192606745831488182616079,
0.890137360295800280281725224499, 2.38022650239251822366217767131, 3.42564324526607843813127527420, 4.27926158868649879108515208909, 5.68640649122396028675197235691, 6.44356807357476632854734062506, 8.042153819591032907143387676673, 9.009985588308919498185416661888, 9.629971972259437566259853266891, 11.05896072684238651553853932524