L(s) = 1 | + (1.83 + 3.17i)2-s + (−2.73 + 4.73i)4-s − 14.7·5-s + (0.242 − 18.5i)7-s + 9.29·8-s + (−27.1 − 46.9i)10-s − 48.5·11-s + (−33.6 − 58.1i)13-s + (59.2 − 33.2i)14-s + (38.9 + 67.4i)16-s + (5.40 + 9.36i)17-s + (67.2 − 116. i)19-s + (40.3 − 69.9i)20-s + (−89.1 − 154. i)22-s − 84.3·23-s + ⋯ |
L(s) = 1 | + (0.648 + 1.12i)2-s + (−0.341 + 0.591i)4-s − 1.32·5-s + (0.0130 − 0.999i)7-s + 0.410·8-s + (−0.857 − 1.48i)10-s − 1.33·11-s + (−0.716 − 1.24i)13-s + (1.13 − 0.633i)14-s + (0.608 + 1.05i)16-s + (0.0771 + 0.133i)17-s + (0.811 − 1.40i)19-s + (0.451 − 0.782i)20-s + (−0.864 − 1.49i)22-s − 0.764·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.741234 - 0.508605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.741234 - 0.508605i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.242 + 18.5i)T \) |
good | 2 | \( 1 + (-1.83 - 3.17i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 14.7T + 125T^{2} \) |
| 11 | \( 1 + 48.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + (33.6 + 58.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-5.40 - 9.36i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-67.2 + 116. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 84.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-55.1 + 95.4i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (75.5 - 130. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (152. - 263. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (127. + 220. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-41.3 + 71.5i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-23.0 - 39.8i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-3.20 - 5.55i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-5.59 + 9.69i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (136. + 235. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-28.7 + 49.7i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 521.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (189. + 327. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-472. - 817. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-411. + 711. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-12.4 + 21.6i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-22.7 + 39.3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.15189290561732368392763768176, −10.87494546529995123676937347550, −10.17396579088773323349091022525, −8.183422489213655585974003031128, −7.63320028979481121205660147143, −6.96140963740026241867493849898, −5.34697960759794133493474757103, −4.55982299113369948198417167970, −3.25236443903682542885557064531, −0.30862868955953955943204841058,
2.06432826268115925858909061781, 3.28389589322139521663268382562, 4.39827629819622861958500357349, 5.48983380408873147817838621390, 7.39193976665583107658557379867, 8.147502708797282280731193386616, 9.592435857742348332680863026323, 10.67043052337721405200312820329, 11.77659794035458822684397977075, 12.00553526231013914566268292478