L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.60 + 0.923i)3-s + (0.499 + 0.866i)4-s + (2.80 + 1.61i)5-s + 1.84·6-s − 0.999i·8-s + (0.207 − 0.358i)9-s + (−1.61 − 2.80i)10-s + (−1.23 − 3.07i)11-s + (−1.60 − 0.923i)12-s + 6.10·13-s − 5.98·15-s + (−0.5 + 0.866i)16-s + (−4.06 − 7.04i)17-s + (−0.358 + 0.207i)18-s + (3.30 − 5.72i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.923 + 0.533i)3-s + (0.249 + 0.433i)4-s + (1.25 + 0.723i)5-s + 0.754·6-s − 0.353i·8-s + (0.0690 − 0.119i)9-s + (−0.511 − 0.886i)10-s + (−0.372 − 0.927i)11-s + (−0.461 − 0.266i)12-s + 1.69·13-s − 1.54·15-s + (−0.125 + 0.216i)16-s + (−0.986 − 1.70i)17-s + (−0.0845 + 0.0488i)18-s + (0.758 − 1.31i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041840275\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041840275\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.23 + 3.07i)T \) |
good | 3 | \( 1 + (1.60 - 0.923i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - 6.10T + 13T^{2} \) |
| 17 | \( 1 + (4.06 + 7.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.30 + 5.72i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.65 + 4.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.32iT - 29T^{2} \) |
| 31 | \( 1 + (2.03 - 1.17i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.64 - 4.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 3.73iT - 43T^{2} \) |
| 47 | \( 1 + (-1.47 - 0.853i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.99 - 3.44i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.82 + 4.51i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.27 + 7.41i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.17 + 2.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + (-5.49 - 9.51i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 + (4.03 + 2.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.82iT - 97T^{2} \) |
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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
−10.00505866388646882369677239201, −9.087263196019856241479650942082, −8.551949862628414365705018844267, −7.05234139690774081055061793625, −6.44246177176939832458737628876, −5.59373957520493202980367666869, −4.80257351484953701222490908275, −3.23030074322864856796942278070, −2.40610177342955345083362651917, −0.72246074377781064028096468838,
1.27955709671131543153445704834, 1.83225686606577345012204487741, 3.86499154449014560000855735699, 5.32192734439695317018514447321, 5.85534996054872131324809834013, 6.36037345506110655769445950145, 7.36201204141011888478993485048, 8.446295763730322380435090802783, 9.052155694281030951674123007131, 9.945283375113309659274069231474