L(s) = 1 | + 2.23i·2-s − 3.00·4-s + (2.5 − 0.866i)7-s − 2.23i·8-s + (−3.87 + 2.23i)11-s + (−1.5 + 0.866i)13-s + (1.93 + 5.59i)14-s − 0.999·16-s + (−3.87 + 6.70i)17-s + (−3 + 1.73i)19-s + (−5.00 − 8.66i)22-s + (3.87 + 2.23i)23-s + (2.5 + 4.33i)25-s + (−1.93 − 3.35i)26-s + (−7.50 + 2.59i)28-s + (3.87 + 2.23i)29-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s + (0.944 − 0.327i)7-s − 0.790i·8-s + (−1.16 + 0.674i)11-s + (−0.416 + 0.240i)13-s + (0.517 + 1.49i)14-s − 0.249·16-s + (−0.939 + 1.62i)17-s + (−0.688 + 0.397i)19-s + (−1.06 − 1.84i)22-s + (0.807 + 0.466i)23-s + (0.5 + 0.866i)25-s + (−0.379 − 0.657i)26-s + (−1.41 + 0.490i)28-s + (0.719 + 0.415i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.156736 - 1.08977i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156736 - 1.08977i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.87 - 2.23i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.87 - 6.70i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.87 - 2.23i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 2.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.87 + 6.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 + 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 + (-3.87 - 2.23i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 - 8.66iT - 61T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 - 8.94iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 + (-3.87 + 6.70i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.74 + 13.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.88846897260432660514470761271, −10.43663946888804254384812168591, −8.935698750877515367880122736475, −8.433374293877543791589859228911, −7.43992375243295659487748064161, −6.98875584569393463054087728828, −5.73543586512284637792277924404, −4.97917073840321649615146251792, −4.12807895727305409895864701553, −2.06452844581411851004323445560,
0.61463765283150756453114400639, 2.38078614325983022667206964627, 2.87862213097586558618610228184, 4.60318469602394998759439561599, 5.00401695479920221390303917803, 6.62111086527781951092278588301, 7.948389739832132563264843100731, 8.743119563101379615575771887603, 9.588731475914183001103564402772, 10.69510514745588797058089419945