L(s) = 1 | − 2-s + (0.5 − 0.866i)3-s + 4-s + (−2.07 + 3.58i)5-s + (−0.5 + 0.866i)6-s + (0.321 − 2.62i)7-s − 8-s + (−0.499 − 0.866i)9-s + (2.07 − 3.58i)10-s + (−0.261 + 0.453i)11-s + (0.5 − 0.866i)12-s + (−3.28 + 1.47i)13-s + (−0.321 + 2.62i)14-s + (2.07 + 3.58i)15-s + 16-s − 2.52·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.926 + 1.60i)5-s + (−0.204 + 0.353i)6-s + (0.121 − 0.992i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.654 − 1.13i)10-s + (−0.0788 + 0.136i)11-s + (0.144 − 0.249i)12-s + (−0.912 + 0.409i)13-s + (−0.0859 + 0.701i)14-s + (0.534 + 0.926i)15-s + 0.250·16-s − 0.611·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0468440 + 0.222014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0468440 + 0.222014i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.321 + 2.62i)T \) |
| 13 | \( 1 + (3.28 - 1.47i)T \) |
good | 5 | \( 1 + (2.07 - 3.58i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.261 - 0.453i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 + (-2.84 - 4.93i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 7.39T + 23T^{2} \) |
| 29 | \( 1 + (1.54 + 2.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.17 + 3.76i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.66T + 37T^{2} \) |
| 41 | \( 1 + (-2.33 - 4.03i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.81 - 8.34i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.58 - 9.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.00192 + 0.00332i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 + (3.00 + 5.20i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.61 - 2.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 6.90i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.99 - 10.3i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.15 + 2.00i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.08T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + (1.31 - 2.28i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.24209202345476339305459751769, −10.17502457793916652852957881378, −9.722771973795253790401832249599, −8.012001420635885243970116505341, −7.73741727185283231521264235342, −6.95839031613649932938896087760, −6.22559885237469504973197117468, −4.21510109415236697247080004812, −3.25834089034242479170059033165, −2.02452400719138416987039130327,
0.14849969111678716171626430907, 2.07672024970241327805232997466, 3.58811405118349492471083770161, 4.90456709403603743253969623147, 5.43556128923171243363015811932, 7.14598100546493120923505239385, 8.127157435875223133219883872603, 8.736011689698316780209846920408, 9.228359891390934113047546182549, 10.19349499539641870214112554897