L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (2 + 1.73i)7-s − 0.999·8-s + (1.5 − 2.59i)9-s + (−1.5 − 2.59i)11-s + 5·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (−1.5 − 2.59i)18-s + (2.5 − 4.33i)19-s − 3·22-s + (−3.5 + 6.06i)23-s + (2.5 − 4.33i)26-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.755 + 0.654i)7-s − 0.353·8-s + (0.5 − 0.866i)9-s + (−0.452 − 0.783i)11-s + 1.38·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + (−0.353 − 0.612i)18-s + (0.573 − 0.993i)19-s − 0.639·22-s + (−0.729 + 1.26i)23-s + (0.490 − 0.849i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41764 - 0.942989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41764 - 0.942989i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + (3.5 - 6.06i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (8 + 13.8i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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
−11.43037689665071350163098355732, −10.68177613968475580988606415535, −9.388118800271014754235695322422, −8.792905082338304325384859022469, −7.60537317777427137050936828727, −6.18504823601605595008371513547, −5.37509321581881459979995080225, −4.07877084750982916932396334106, −2.96050640500358563979940233413, −1.31034123696361728243552015324,
1.84383325856557009391926571129, 3.84342406883718544027527528705, 4.67526715532093570447953523581, 5.77708077183459598292002559870, 6.95807417259234074157050225892, 7.897430831147115771947835116270, 8.432844752820524408629777323891, 9.990005599916454234639780513615, 10.67423866995848936287344453353, 11.66323269048498479280085956263