L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.44 − 1.41i)5-s + (−2 − 1.73i)7-s − 2.82i·8-s + (1.99 + 3.46i)10-s + (−1.22 − 2.12i)11-s − 13-s + (1.22 + 3.53i)14-s + (−2.00 + 3.46i)16-s + (−1.22 + 0.707i)17-s + (6 + 3.46i)19-s − 5.65i·20-s + 3.46i·22-s + (1.22 − 2.12i)23-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.09 − 0.632i)5-s + (−0.755 − 0.654i)7-s − 0.999i·8-s + (0.632 + 1.09i)10-s + (−0.369 − 0.639i)11-s − 0.277·13-s + (0.327 + 0.944i)14-s + (−0.500 + 0.866i)16-s + (−0.297 + 0.171i)17-s + (1.37 + 0.794i)19-s − 1.26i·20-s + 0.738i·22-s + (0.255 − 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.135109 + 0.118984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.135109 + 0.118984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.22 + 2.12i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (1.22 - 0.707i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 + (4.5 - 2.59i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + 8.66iT - 43T^{2} \) |
| 47 | \( 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.57 - 4.94i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.67 + 6.36i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.5 + 2.59i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.5 + 7.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.07i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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
−10.56789003183533916444580621442, −9.660960244036871498490535212884, −8.863768331050593891855680873156, −8.011807842035768831251053811159, −7.42289137091290613774990391408, −6.46470033675729656076424153023, −4.96138991969217475995384331888, −3.72389265616846728367821978826, −3.11574873380778136081104632195, −1.17345610309763894201329371499,
0.13679411927205685244502742793, 2.32111898726139404263118187507, 3.40481054725593714985678936477, 4.91082626868170050263950991477, 5.91756337016283453576551807467, 7.06627179263821952092969281116, 7.39549580121840840273123122729, 8.341459911460738194130290904497, 9.444859013154335709619176815272, 9.801889779536709560834962883439