L(s) = 1 | + (1.33 − 2.30i)3-s + 3.16i·5-s + (−0.866 + 0.5i)7-s + (−2.03 − 3.53i)9-s + (−5.14 − 2.97i)11-s + (−0.0766 − 3.60i)13-s + (7.28 + 4.20i)15-s + (−1.34 − 2.33i)17-s + (−1.69 + 0.978i)19-s + 2.66i·21-s + (1.36 − 2.36i)23-s − 4.99·25-s − 2.86·27-s + (2.99 − 5.19i)29-s − 1.15i·31-s + ⋯ |
L(s) = 1 | + (0.767 − 1.33i)3-s + 1.41i·5-s + (−0.327 + 0.188i)7-s + (−0.679 − 1.17i)9-s + (−1.55 − 0.895i)11-s + (−0.0212 − 0.999i)13-s + (1.88 + 1.08i)15-s + (−0.327 − 0.567i)17-s + (−0.388 + 0.224i)19-s + 0.580i·21-s + (0.284 − 0.492i)23-s − 0.999·25-s − 0.551·27-s + (0.556 − 0.964i)29-s − 0.206i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.147451246\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.147451246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (0.0766 + 3.60i)T \) |
good | 3 | \( 1 + (-1.33 + 2.30i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 3.16iT - 5T^{2} \) |
| 11 | \( 1 + (5.14 + 2.97i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.34 + 2.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.69 - 0.978i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 2.36i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.99 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.15iT - 31T^{2} \) |
| 37 | \( 1 + (5.63 + 3.25i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.23 + 1.86i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.49 + 6.05i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.456iT - 47T^{2} \) |
| 53 | \( 1 - 0.399T + 53T^{2} \) |
| 59 | \( 1 + (4.16 - 2.40i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.578 - 1.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.43 - 3.13i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.90 - 2.25i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8.30iT - 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 83 | \( 1 - 6.19iT - 83T^{2} \) |
| 89 | \( 1 + (-3.08 - 1.78i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.96 - 1.71i)T + (48.5 - 84.0i)T^{2} \) |
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
−8.898517400359097339825185831452, −8.136498901342510492531127580845, −7.62153030004322132799323302285, −6.86060612027684887012209125062, −6.19674206052764417602942979820, −5.28123826388745418002015820047, −3.47037753593701446030264579544, −2.75834331478044198002687550151, −2.30637936926682309877175565963, −0.38171625029470782758074748190,
1.77033382219601657973763286313, 3.00447623593290380708445969551, 4.06543147359003012759917844834, 4.85884280368581702232960899939, 5.10924494643204164436581998612, 6.61719093355053506507571360442, 7.75002832660779215714773218467, 8.583537251322333200041246319966, 8.986393916857293844587348842506, 9.817112646269171695205060942818