L(s) = 1 | + (0.347 − 0.601i)5-s + 0.305·7-s − 4.82·11-s + (−0.5 − 0.866i)13-s + (−3.75 + 6.51i)17-s + (3.06 + 3.10i)19-s + (0.347 + 0.601i)23-s + (2.25 + 3.91i)25-s + (5.06 + 8.77i)29-s − 1.82·31-s + (0.106 − 0.183i)35-s + 6.51·37-s + (2.69 − 4.66i)41-s + (−1.84 + 3.19i)43-s + (−3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.155 − 0.269i)5-s + 0.115·7-s − 1.45·11-s + (−0.138 − 0.240i)13-s + (−0.911 + 1.57i)17-s + (0.702 + 0.711i)19-s + (0.0724 + 0.125i)23-s + (0.451 + 0.782i)25-s + (0.940 + 1.62i)29-s − 0.327·31-s + (0.0179 − 0.0310i)35-s + 1.07·37-s + (0.420 − 0.728i)41-s + (−0.281 + 0.487i)43-s + (−0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0547 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0547 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.143204636\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143204636\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.06 - 3.10i)T \) |
good | 5 | \( 1 + (-0.347 + 0.601i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 0.305T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.75 - 6.51i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.347 - 0.601i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.06 - 8.77i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 - 6.51T + 37T^{2} \) |
| 41 | \( 1 + (-2.69 + 4.66i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.84 - 3.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.71 - 4.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.04 - 3.53i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.194 + 0.337i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.91 - 6.77i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.45 + 9.44i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.19 + 3.80i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.21 - 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.739T + 83T^{2} \) |
| 89 | \( 1 + (0.411 + 0.712i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.45 - 9.44i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.861890324982280275189190468406, −8.874576128654966283289485470097, −8.191855120054440758240599646379, −7.48858163325736010576962980179, −6.46628809403850662090643797355, −5.49862487186304811298833459574, −4.90078291673612469805789538801, −3.71096428534882567228945134019, −2.65091461773479678851816683072, −1.42564037333114073648210165034,
0.46707141351337917191615402969, 2.40821147042629645041435822305, 2.89965496176559572892509945095, 4.53342190598774434080154771440, 5.01941661437313986176489374019, 6.14881390644942302402376883029, 6.99511085983117686360823090769, 7.75846745321920192609229206641, 8.524598924143990088850319151014, 9.597740319249801648853037428736