L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.866 − 0.5i)11-s + (−0.965 + 0.258i)12-s + (−0.258 − 0.965i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.965 + 0.258i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.866 − 0.5i)11-s + (−0.965 + 0.258i)12-s + (−0.258 − 0.965i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5360589805\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5360589805\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.258 - 0.965i)T \) |
| 7 | \( 1 + (0.965 + 0.258i)T \) |
good | 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 19 | \( 1 - 1.41iT - T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1 - i)T + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - iT - T^{2} \) |
| 73 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
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
−10.29052042505432216499181585858, −9.225671791322955044230002701626, −8.056858583939199620812741119000, −7.47725896407060140727916834688, −6.67918622897795451968890610879, −5.96373649234806029161281045835, −5.65728314873704411409365513059, −3.74032679380075073628163374079, −2.52940146873883400664021505313, −1.13467938091675404829471435678,
0.830967595032198311776517594613, 2.40700394541610343815090493996, 3.74607989247494386457915952986, 4.51945400730650683252336199682, 5.59926086924605759014766324345, 6.65196359948218367765170621697, 7.21413595436967274331725204506, 8.766964038585687432924747002678, 9.206219254971462568177729916039, 9.640487969231627730219400606310