L(s) = 1 | − 3.53i·2-s − 8.46·4-s − 6.19·5-s + 2.58i·7-s + 15.7i·8-s + 21.8i·10-s + (−6.73 − 8.69i)11-s − 23.7i·13-s + 9.12·14-s + 21.7·16-s + 12.2i·17-s − 3.27i·19-s + 52.4·20-s + (−30.7 + 23.7i)22-s + 14.3·23-s + ⋯ |
L(s) = 1 | − 1.76i·2-s − 2.11·4-s − 1.23·5-s + 0.369i·7-s + 1.97i·8-s + 2.18i·10-s + (−0.612 − 0.790i)11-s − 1.82i·13-s + 0.651·14-s + 1.36·16-s + 0.719i·17-s − 0.172i·19-s + 2.62·20-s + (−1.39 + 1.08i)22-s + 0.623·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.253004 + 0.515702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.253004 + 0.515702i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (6.73 + 8.69i)T \) |
good | 2 | \( 1 + 3.53iT - 4T^{2} \) |
| 5 | \( 1 + 6.19T + 25T^{2} \) |
| 7 | \( 1 - 2.58iT - 49T^{2} \) |
| 13 | \( 1 + 23.7iT - 169T^{2} \) |
| 17 | \( 1 - 12.2iT - 289T^{2} \) |
| 19 | \( 1 + 3.27iT - 361T^{2} \) |
| 23 | \( 1 - 14.3T + 529T^{2} \) |
| 29 | \( 1 + 38.5iT - 841T^{2} \) |
| 31 | \( 1 + 11.1T + 961T^{2} \) |
| 37 | \( 1 + 12.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.38iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 12.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 21.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34T + 4.48e3T^{2} \) |
| 71 | \( 1 - 69.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 39.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 71.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 166.T + 9.40e3T^{2} \) |
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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
−12.71692555950177422458913399980, −11.81257913234195749655003323529, −10.95298681321864520106781771119, −10.20731013450066108123096572831, −8.682455913229222244646871275212, −7.88008278307384103051188761633, −5.42064793063244208870924899004, −3.86702034632014732685982197092, −2.83631151900026662801285429690, −0.44472646266925324372337709955,
4.08420637090353443881296211239, 5.03960608206506129436887541313, 6.88145112202815231241828686318, 7.31824401889604623559207793987, 8.475701547038382132766080744380, 9.526784787038551729816924164938, 11.23584574537176635156005456843, 12.46060122650998994524639631909, 13.73998403148635193034467838042, 14.59320987962549858920434904606