L(s) = 1 | + (2.09 + 1.52i)2-s + (−1.46 − 0.921i)3-s + (1.44 + 4.45i)4-s + (0.587 + 0.809i)5-s + (−1.66 − 4.15i)6-s + (0.326 − 0.106i)7-s + (−2.14 + 6.61i)8-s + (1.30 + 2.70i)9-s + 2.58i·10-s + (−2.88 − 1.63i)11-s + (1.98 − 7.87i)12-s + (3.76 − 5.17i)13-s + (0.845 + 0.274i)14-s + (−0.116 − 1.72i)15-s + (−6.96 + 5.05i)16-s + (0.839 − 0.610i)17-s + ⋯ |
L(s) = 1 | + (1.47 + 1.07i)2-s + (−0.846 − 0.532i)3-s + (0.724 + 2.22i)4-s + (0.262 + 0.361i)5-s + (−0.680 − 1.69i)6-s + (0.123 − 0.0401i)7-s + (−0.759 + 2.33i)8-s + (0.433 + 0.901i)9-s + 0.817i·10-s + (−0.869 − 0.494i)11-s + (0.573 − 2.27i)12-s + (1.04 − 1.43i)13-s + (0.225 + 0.0733i)14-s + (−0.0299 − 0.446i)15-s + (−1.74 + 1.26i)16-s + (0.203 − 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48973 + 1.21134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48973 + 1.21134i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.46 + 0.921i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 + (2.88 + 1.63i)T \) |
good | 2 | \( 1 + (-2.09 - 1.52i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.326 + 0.106i)T + (5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.76 + 5.17i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.839 + 0.610i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (5.97 + 1.94i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 1.43iT - 23T^{2} \) |
| 29 | \( 1 + (1.36 + 4.20i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.974 - 0.708i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.0967 - 0.297i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.390 - 1.20i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 + (-8.00 - 2.60i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.614 - 0.846i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (7.13 - 2.31i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.01 - 2.77i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 7.66T + 67T^{2} \) |
| 71 | \( 1 + (5.34 + 7.36i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.99 - 1.29i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.88 - 3.96i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 7.77i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 4.84iT - 89T^{2} \) |
| 97 | \( 1 + (8.10 + 5.88i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.18988669947504915205238729775, −12.56867281651281983494693091895, −11.29685931035446856697772683632, −10.55200843999677248983979793529, −8.234246603920811643224828128568, −7.50690752182186731216639760921, −6.19166956346441320395681494274, −5.79128136682384871454544781687, −4.59469965393680031270854773001, −2.94295102760184084693258585261,
1.86701098500572039145329513086, 3.83230132057226628434983823241, 4.67189705771430828887058452492, 5.68662791682505454758204990576, 6.63008253502389407698460454953, 8.954710054599300923824239530581, 10.25875030730816088867488462353, 10.80317806792880802591100083197, 11.77846985239259332140302398389, 12.52633604297344832297345237285