L(s) = 1 | + (0.866 − 0.5i)2-s + (2.59 + 1.5i)3-s + (0.499 − 0.866i)4-s + 3·6-s + (−2.59 + 0.5i)7-s − 0.999i·8-s + (3 + 5.19i)9-s + (1 − 1.73i)11-s + (2.59 − 1.50i)12-s + (−2 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (3.46 + 2i)17-s + (5.19 + 3i)18-s + (−3 − 5.19i)19-s + (−7.5 − 2.59i)21-s − 1.99i·22-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (1.49 + 0.866i)3-s + (0.249 − 0.433i)4-s + 1.22·6-s + (−0.981 + 0.188i)7-s − 0.353i·8-s + (1 + 1.73i)9-s + (0.301 − 0.522i)11-s + (0.749 − 0.433i)12-s + (−0.534 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.840 + 0.485i)17-s + (1.22 + 0.707i)18-s + (−0.688 − 1.19i)19-s + (−1.63 − 0.566i)21-s − 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.65573 + 0.248637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65573 + 0.248637i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.59 - 0.5i)T \) |
good | 3 | \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.73 + i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 - 4.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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
−11.45830394211616787958044639117, −10.39367348336138463390927457975, −9.648154246753927218821806552510, −8.973768486050658846502796731174, −7.978059733327995750764723036938, −6.65823937061205678697031699098, −5.37638654993758250888226586360, −3.97614409054278345314546042821, −3.38883812772728956687879709510, −2.29224105119528980137942585785,
1.93660913089026135260702039614, 3.22269022082109992907394517546, 3.99346796826565650026334032043, 5.83062831076453301823990440856, 6.90954733835177592113914769354, 7.51263425677502025768590569392, 8.480947616722484739014020654246, 9.399565888172569240444973827644, 10.30657668886049923464921036324, 12.10337286420655333028359054383