L(s) = 1 | + (0.739 − 2.27i)2-s + (0.698 + 1.58i)3-s + (−3.01 − 2.19i)4-s + (1.17 − 0.382i)5-s + (4.12 − 0.419i)6-s + (1.66 − 2.28i)7-s + (−3.35 + 2.43i)8-s + (−2.02 + 2.21i)9-s − 2.96i·10-s + (1.36 − 6.31i)12-s + (3.18 + 1.03i)13-s + (−3.98 − 5.47i)14-s + (1.43 + 1.60i)15-s + (0.761 + 2.34i)16-s + (−1.28 − 3.94i)17-s + (3.54 + 6.24i)18-s + ⋯ |
L(s) = 1 | + (0.523 − 1.61i)2-s + (0.403 + 0.914i)3-s + (−1.50 − 1.09i)4-s + (0.527 − 0.171i)5-s + (1.68 − 0.171i)6-s + (0.628 − 0.864i)7-s + (−1.18 + 0.861i)8-s + (−0.674 + 0.738i)9-s − 0.938i·10-s + (0.394 − 1.82i)12-s + (0.882 + 0.286i)13-s + (−1.06 − 1.46i)14-s + (0.369 + 0.413i)15-s + (0.190 + 0.585i)16-s + (−0.310 − 0.956i)17-s + (0.836 + 1.47i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31125 - 1.60214i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31125 - 1.60214i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.698 - 1.58i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.739 + 2.27i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (-1.17 + 0.382i)T + (4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.66 + 2.28i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-3.18 - 1.03i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.28 + 3.94i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.43 - 1.98i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + (4.39 + 3.19i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.0606 + 0.186i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.46 - 6.15i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.81 - 4.22i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + (-1.99 - 2.73i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.75 - 2.19i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (7.42 - 10.2i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.29 + 1.39i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + (7.93 - 2.57i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.76 - 6.55i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.98 - 2.27i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.58 + 14.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 9.58iT - 89T^{2} \) |
| 97 | \( 1 + (-0.700 + 2.15i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.32672348208814745852428759303, −10.31599677134328682592452896594, −9.723583957622793602351348500829, −8.915364785830018139432038115560, −7.66263410212715963165161439942, −5.76922931465957085944843646279, −4.65652146396387447284966329323, −3.99783000659719557964369124221, −2.85210409711616780458352137091, −1.46895315135471035828532844763,
2.06230628690370889400256283028, 3.77925298277469853348531785149, 5.38038928216800630408648349285, 5.97148976628766330919293075652, 6.81074950369894397030820333309, 7.84155546602349444898824944478, 8.518560418519149338278908217254, 9.210092067756248545402740947120, 10.90572673238907540739235115125, 12.05689375317884268142817438226