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 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.640487969231627730219400606310, −9.206219254971462568177729916039, −8.766964038585687432924747002678, −7.21413595436967274331725204506, −6.65196359948218367765170621697, −5.59926086924605759014766324345, −4.51945400730650683252336199682, −3.74607989247494386457915952986, −2.40700394541610343815090493996, −0.830967595032198311776517594613,
1.13467938091675404829471435678, 2.52940146873883400664021505313, 3.74032679380075073628163374079, 5.65728314873704411409365513059, 5.96373649234806029161281045835, 6.67918622897795451968890610879, 7.47725896407060140727916834688, 8.056858583939199620812741119000, 9.225671791322955044230002701626, 10.29052042505432216499181585858