L(s) = 1 | + (−1.68 − 0.388i)3-s + (−0.538 + 0.932i)5-s + (−4.07 + 2.35i)7-s + (2.69 + 1.31i)9-s + (4.28 − 2.47i)11-s + (1.05 + 0.606i)13-s + (1.27 − 1.36i)15-s + 3.12i·17-s − 2.05·19-s + (7.78 − 2.38i)21-s + (−1.71 + 2.97i)23-s + (1.91 + 3.32i)25-s + (−4.04 − 3.26i)27-s + (−0.723 − 1.25i)29-s + (−6.45 − 3.72i)31-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.224i)3-s + (−0.240 + 0.417i)5-s + (−1.53 + 0.888i)7-s + (0.899 + 0.437i)9-s + (1.29 − 0.746i)11-s + (0.291 + 0.168i)13-s + (0.328 − 0.352i)15-s + 0.756i·17-s − 0.472·19-s + (1.69 − 0.520i)21-s + (−0.358 + 0.620i)23-s + (0.383 + 0.665i)25-s + (−0.777 − 0.628i)27-s + (−0.134 − 0.232i)29-s + (−1.15 − 0.669i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03061015182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03061015182\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 + 0.388i)T \) |
good | 5 | \( 1 + (0.538 - 0.932i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (4.07 - 2.35i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.28 + 2.47i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.05 - 0.606i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.12iT - 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + (1.71 - 2.97i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.723 + 1.25i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.45 + 3.72i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 9.04iT - 37T^{2} \) |
| 41 | \( 1 + (8.75 + 5.05i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0592 + 0.102i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.98 - 3.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.401T + 53T^{2} \) |
| 59 | \( 1 + (8.97 + 5.18i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.7 - 6.77i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.94 + 8.55i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.327T + 73T^{2} \) |
| 79 | \( 1 + (-8.98 + 5.18i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (8.21 - 4.74i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9.80iT - 89T^{2} \) |
| 97 | \( 1 + (-2.79 - 4.84i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.329092639202499500651549234462, −8.948739998597687664633029564588, −7.56911196961615469694956436454, −6.66598679952579913732650450466, −6.11965152868140138719250345651, −5.59274612677209319140217790038, −4.00130351112292366264243667554, −3.34795809918134196577611183661, −1.79084821389106574908445494558, −0.01650091153897774741977809397,
1.27564129661363974772587432795, 3.28643246483792417625114378152, 4.15832133824561517113630641731, 4.84686479163293472232503111770, 6.20393000355387482225427409332, 6.67579238508803405796110593228, 7.30638492929869297337250548875, 8.723399882229225575489659290477, 9.540703252346670055738746444390, 10.11566120915639572499196285562