L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.866 + 0.499i)6-s + (−0.465 − 0.807i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (1.45 + 0.841i)11-s + 0.999i·12-s + (3.08 + 1.86i)13-s − 0.931·14-s + (−0.5 + 0.866i)16-s + (2.21 − 1.27i)17-s + 0.999·18-s + (−2.27 + 1.31i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.353 + 0.204i)6-s + (−0.176 − 0.305i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.439 + 0.253i)11-s + 0.288i·12-s + (0.856 + 0.516i)13-s − 0.249·14-s + (−0.125 + 0.216i)16-s + (0.536 − 0.309i)17-s + 0.235·18-s + (−0.522 + 0.301i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790761935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790761935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.08 - 1.86i)T \) |
good | 7 | \( 1 + (0.465 + 0.807i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.45 - 0.841i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.21 + 1.27i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.27 - 1.31i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.59 - 3.22i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.69 - 2.93i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.29iT - 31T^{2} \) |
| 37 | \( 1 + (-4.57 + 7.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 0.587i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.55 + 1.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + 5.36iT - 53T^{2} \) |
| 59 | \( 1 + (-3.44 + 1.98i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.36 - 7.55i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.59 + 4.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.40 + 1.96i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.26T + 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 - 3.84T + 83T^{2} \) |
| 89 | \( 1 + (11.6 + 6.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.04 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.181346798017085834075895676221, −8.431055657856308599039676624020, −7.23000873931761304565782044209, −6.70727107310102605985730022938, −5.75200781472121542976286604915, −5.01143241993130496753233497541, −4.00663659586171735258692558557, −3.27104090659391209746382636436, −1.89838215918216042582059927383, −0.945493926788141734074948948399,
0.875667863558646217505392187554, 2.65964565107735102766456039420, 3.71080955485916705948140575743, 4.47340256167118152627740976735, 5.45143037229368903526513433184, 6.13175842779252411192149715289, 6.64280735666956468493911063202, 7.76211836805134854817036345541, 8.437910249232792018839914270360, 9.245865031661704761380729612127