L(s) = 1 | − i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.866 + 0.5i)7-s + i·8-s + (0.866 + 0.5i)10-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s − 0.999i·15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3844 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3844 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4521546607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4521546607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.483910304594577036641104570830, −8.358695895237194425601934561993, −7.45422293492670536248461065852, −6.53521263904980482599127701363, −5.96862862527537985113554570811, −4.92550396290911907662507257298, −4.35524717144179883105943812572, −3.39774603499866286587083693913, −2.81106218925979491453209574361, −1.56411518011763887556406871093,
0.37160899502974067805017692342, 1.04727460259040058055317214806, 3.28364262987471721160660919565, 3.84359044325451143422545770489, 5.00935076514477544437205215259, 5.44768296017778034406297917890, 6.24702998427685362018194922820, 6.87708797455133627139975859209, 7.50164914295475954492272308563, 8.200952674362229241628139473296