L(s) = 1 | + (2.05 − 1.18i)2-s + (1.82 − 3.15i)4-s + (1.46 + 1.69i)5-s + (−2.03 − 1.68i)7-s − 3.91i·8-s + (5.01 + 1.74i)10-s + (1.80 − 3.13i)11-s + 3.36i·13-s + (−6.19 − 1.04i)14-s + (−1 − 1.73i)16-s + (−4.71 − 2.72i)17-s + (2.32 + 4.02i)19-s + (8.00 − 1.53i)20-s − 8.58i·22-s + (−0.599 + 0.346i)23-s + ⋯ |
L(s) = 1 | + (1.45 − 0.840i)2-s + (0.911 − 1.57i)4-s + (0.654 + 0.756i)5-s + (−0.771 − 0.636i)7-s − 1.38i·8-s + (1.58 + 0.551i)10-s + (0.544 − 0.943i)11-s + 0.934i·13-s + (−1.65 − 0.278i)14-s + (−0.250 − 0.433i)16-s + (−1.14 − 0.660i)17-s + (0.532 + 0.923i)19-s + (1.79 − 0.343i)20-s − 1.83i·22-s + (−0.125 + 0.0722i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.41055 - 1.45803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.41055 - 1.45803i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.46 - 1.69i)T \) |
| 7 | \( 1 + (2.03 + 1.68i)T \) |
good | 2 | \( 1 + (-2.05 + 1.18i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.80 + 3.13i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.36iT - 13T^{2} \) |
| 17 | \( 1 + (4.71 + 2.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.32 - 4.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.599 - 0.346i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.51 - 2.60i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 8.91iT - 43T^{2} \) |
| 47 | \( 1 + (-3.38 + 1.95i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.57 + 3.21i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.388 + 0.673i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.82 + 4.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.9 - 6.90i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.00T + 71T^{2} \) |
| 73 | \( 1 + (-1.31 - 0.760i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.96 + 8.60i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.76iT - 83T^{2} \) |
| 89 | \( 1 + (1.80 + 3.13i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 12.2iT - 97T^{2} \) |
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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
−11.44214405502406382537615910091, −10.95463417739136064215379543051, −9.984640450241120282824311896387, −9.063426090806387771154461902534, −7.08723531154173395041632042693, −6.35323270471471025138771553477, −5.43206091858885998864411393871, −3.96444781411114473339492533870, −3.26669718652410331720638951487, −1.90764107240463486536423906909,
2.40847477748712062714605006787, 3.89073055860627181531101698300, 4.95979814736277836275683699178, 5.81589634377984505671434171862, 6.56676266030034882356629554514, 7.65482558255843102826096664409, 9.008469385834205693370183882610, 9.716313992474551188942899021823, 11.24927266439567734681240763754, 12.51405342695737996084449580566