L(s) = 1 | + (0.5 + 1.86i)3-s + (−0.133 − 2.23i)5-s + (−0.866 − 2.5i)7-s + (−0.633 + 0.366i)9-s + (0.366 − 0.633i)11-s + (2 − 2i)13-s + (4.09 − 1.36i)15-s + (1 − 0.267i)17-s + (−1.36 − 2.36i)19-s + (4.23 − 2.86i)21-s + (1.86 − 6.96i)23-s + (−4.96 + 0.598i)25-s + (3.09 + 3.09i)27-s + 3i·29-s + (−0.464 − 0.267i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 1.07i)3-s + (−0.0599 − 0.998i)5-s + (−0.327 − 0.944i)7-s + (−0.211 + 0.122i)9-s + (0.110 − 0.191i)11-s + (0.554 − 0.554i)13-s + (1.05 − 0.352i)15-s + (0.242 − 0.0649i)17-s + (−0.313 − 0.542i)19-s + (0.923 − 0.625i)21-s + (0.389 − 1.45i)23-s + (−0.992 + 0.119i)25-s + (0.596 + 0.596i)27-s + 0.557i·29-s + (−0.0833 − 0.0481i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46736 - 0.419564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46736 - 0.419564i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.133 + 2.23i)T \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 3 | \( 1 + (-0.5 - 1.86i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1 + 0.267i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.36 + 2.36i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.86 + 6.96i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.73 - 1.26i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.464iT - 41T^{2} \) |
| 43 | \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.169 - 0.633i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.09 + 1.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.33 - 4.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.303 - 1.13i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 + (-0.928 - 3.46i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (5.83 - 3.36i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.09 + 3.09i)T - 83iT^{2} \) |
| 89 | \( 1 + (8.33 + 14.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.92 + 7.92i)T + 97iT^{2} \) |
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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
−10.52778810044323177770578778338, −9.826844302292207034364829750791, −8.979993543085423816835341089565, −8.322269558593354684035253872382, −7.15527135632301220956647719078, −5.95722975110971380557796720793, −4.70750490172076158066300270099, −4.17277695570369197425993102220, −3.08586664357771231855932281104, −0.919768922306661910640116325200,
1.73556471878754538975600940548, 2.71980241819848399931898225184, 3.91006254439122032185517464871, 5.66895033600833930747938904689, 6.41338532177598580278298978815, 7.24811577245422869171332855600, 7.980021468989728217613862527843, 9.025651123465120256463663574949, 9.905790996090266697897797915546, 10.98045839201117166889261629433