L(s) = 1 | + (1.30 − 0.951i)2-s + (−0.309 + 0.951i)3-s + (0.190 − 0.587i)4-s + 2.23·5-s + (0.499 + 1.53i)6-s − 7-s + (0.690 + 2.12i)8-s + (−0.809 − 0.587i)9-s + (2.92 − 2.12i)10-s + (0.618 − 0.449i)11-s + (0.5 + 0.363i)12-s + (2.80 + 2.04i)13-s + (−1.30 + 0.951i)14-s + (−0.690 + 2.12i)15-s + (3.92 + 2.85i)16-s + (0.454 + 1.40i)17-s + ⋯ |
L(s) = 1 | + (0.925 − 0.672i)2-s + (−0.178 + 0.549i)3-s + (0.0954 − 0.293i)4-s + 0.999·5-s + (0.204 + 0.628i)6-s − 0.377·7-s + (0.244 + 0.751i)8-s + (−0.269 − 0.195i)9-s + (0.925 − 0.672i)10-s + (0.186 − 0.135i)11-s + (0.144 + 0.104i)12-s + (0.779 + 0.566i)13-s + (−0.349 + 0.254i)14-s + (−0.178 + 0.549i)15-s + (0.981 + 0.713i)16-s + (0.110 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.47368 + 0.155631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47368 + 0.155631i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 + (-1.30 + 0.951i)T + (0.618 - 1.90i)T^{2} \) |
| 11 | \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.80 - 2.04i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.454 - 1.40i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (1.11 + 3.44i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.309 + 0.224i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.07 - 6.37i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.19 + 3.66i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.927 + 0.673i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.11 + 2.99i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (-3.70 + 11.4i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.26 + 3.88i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.35 + 6.06i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.236 - 0.171i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.78 + 5.48i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.454 + 1.40i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.3 - 8.28i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.33 - 7.19i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4 + 12.3i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.61 - 2.62i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.54 + 10.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.93792276096069653142205904935, −10.28835532079281482478015036278, −9.192582154092166529461593558606, −8.567878420167945118961505142994, −6.91771776750852167465237574510, −5.91417708724242520864586791079, −5.12486970892762200839904860534, −4.05728632087198794572034964292, −3.12673878712068688196868780349, −1.85836653948862329741753387911,
1.36045817475820196998737946707, 3.03246709456858498475918707015, 4.37422959629813724442148635519, 5.62259359932216222501328682576, 6.02197424421362615696617875634, 6.84250419727852991433220720269, 7.84431998514043485500010396307, 9.089868481589282570386762131542, 10.00282249621521797849903070823, 10.76688449871811639144483992321