L(s) = 1 | + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−0.448 − 1.67i)18-s + (−1.22 − 1.22i)22-s + (−1.67 − 0.448i)23-s + i·29-s − 2·36-s + (−0.448 + 1.67i)37-s + (1.22 − 1.22i)43-s + (−1.73 + 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
L(s) = 1 | + (0.448 − 1.67i)2-s + (−1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (0.866 − 0.5i)9-s + (0.5 − 0.866i)11-s + (0.500 + 0.866i)16-s + (−0.448 − 1.67i)18-s + (−1.22 − 1.22i)22-s + (−1.67 − 0.448i)23-s + i·29-s − 2·36-s + (−0.448 + 1.67i)37-s + (1.22 − 1.22i)43-s + (−1.73 + 0.999i)44-s + (−1.50 + 2.59i)46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.253404666\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253404666\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 3 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.857573619183519056394542995083, −9.070432743677671488912801103881, −8.258017116831273025968414835014, −6.94864420413715345271757100080, −5.99340255110725250179377493561, −4.90769542363948010905261956956, −3.96647107914123006314442429121, −3.41650230896378843816209700367, −2.17338782806571675765020096539, −1.06208500442160328966174418076,
2.02665743597294185186957471526, 3.96349469404813819988081851981, 4.36941610548638409971685415239, 5.41763965168010985641671744212, 6.18397927557874651690128700270, 7.04611701723352898002165096824, 7.62828153376467313284627957721, 8.259569190457153693089973165668, 9.387283160420813157629406358735, 9.898463940325700867872528587196