L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.35 + 0.779i)3-s + (0.499 + 0.866i)4-s + (−0.882 − 0.509i)5-s − 1.55·6-s + 0.999i·8-s + (−0.284 + 0.492i)9-s + (−0.509 − 0.882i)10-s + (−2.58 − 2.08i)11-s + (−1.35 − 0.779i)12-s − 0.167·13-s + 1.58·15-s + (−0.5 + 0.866i)16-s + (−1.47 − 2.55i)17-s + (−0.492 + 0.284i)18-s + (−0.155 + 0.269i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.779 + 0.450i)3-s + (0.249 + 0.433i)4-s + (−0.394 − 0.227i)5-s − 0.636·6-s + 0.353i·8-s + (−0.0947 + 0.164i)9-s + (−0.161 − 0.279i)10-s + (−0.778 − 0.627i)11-s + (−0.389 − 0.225i)12-s − 0.0463·13-s + 0.410·15-s + (−0.125 + 0.216i)16-s + (−0.358 − 0.620i)17-s + (−0.116 + 0.0670i)18-s + (−0.0357 + 0.0619i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0910 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0910 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4338814588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4338814588\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.58 + 2.08i)T \) |
good | 3 | \( 1 + (1.35 - 0.779i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.882 + 0.509i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 0.167T + 13T^{2} \) |
| 17 | \( 1 + (1.47 + 2.55i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.155 - 0.269i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.237 + 0.411i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 1.89iT - 29T^{2} \) |
| 31 | \( 1 + (-2.20 + 1.27i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.04 - 5.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-3.28 - 1.89i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.21 + 7.29i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.40 - 3.12i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.93 + 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.19 - 8.99i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + (4.85 + 8.40i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.06 + 3.50i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + (5.97 + 3.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.893929359680316579664988247339, −8.547409199397271948907679554956, −8.056018418343615229365579079438, −6.97565864029934316327796886656, −6.07092645528431050943992409229, −5.22885732274317225363376326875, −4.69307323757684689697802621811, −3.61635180947655728706912320060, −2.42003462193485422101926746770, −0.16811917301608920703580924438,
1.54815776409799270004565621062, 2.87950316092447523025017121107, 3.94332885489629942009394777069, 5.01091927756277459226444088953, 5.74832494337798475292249054251, 6.67891448696588760077841031590, 7.31531744285157343418354251564, 8.345146325446991458347773553825, 9.471208414710130075944067929727, 10.38734108599370766158984844661