L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 3i·5-s + (0.866 + 0.499i)6-s + (1.73 + i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)10-s + (−5.19 + 3i)11-s − 0.999·12-s − 1.99·14-s + (2.59 − 1.5i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 1.34i·5-s + (0.353 + 0.204i)6-s + (0.654 + 0.377i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (−1.56 + 0.904i)11-s − 0.288·12-s − 0.534·14-s + (0.670 − 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3045994253\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3045994253\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (5.19 - 3i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.73 + i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (-6.06 + 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.19 + 3i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (15.5 - 9i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12.1 + 7i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.40196732341519270587379814448, −9.800936314926166514588750353111, −8.384454774366767922408590003084, −7.906408900987525058772923323388, −7.10144822079051072817305174148, −6.38303986898124058122764328178, −5.52468659631156744812410746725, −4.42886998164188985735267054985, −2.63167674060205042562390229424, −2.06291804708833279657233554543,
0.17037659392727290547830048564, 1.49914467598205561085488530505, 3.00962532090607109482197805874, 4.28310511879154385523465420332, 5.09798098983953561566367651306, 5.78043212763557685925931292237, 7.30389471747416490914081847080, 8.172270356614773359007508480298, 8.637202112656279381407238922704, 9.513571466345099060427374749447