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
\(L(1)\) |
\(\approx\) |
\(2.565684216\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.565684216\) |
\(L(\frac{3}{2})\) |
|
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
−9.515562279736488634383514717889, −8.661255836998725653733120618501, −8.025817063025445917309079587093, −7.83308657245303455594235337399, −6.42516692632423077067847304666, −5.14240966341129951683895805794, −4.77637643481416267951769727825, −3.24624420024989939488456106480, −2.43982594302024268276537323393, −1.19782497250056022933582039932,
1.49456435130998762810398497819, 2.54229641485953976338523275006, 3.73727083959708087243568470369, 4.56869018223684686608555217056, 5.29575261694105862131522778735, 7.00581379664307668131783206471, 7.47685404781366087309586424515, 8.198847968332721152890558212714, 8.798417715853049455423955467993, 10.12032537472265553355846284274