L(s) = 1 | + (0.5 − 0.866i)2-s + (−1.18 − 1.26i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 1.65i)5-s + (−1.68 + 0.396i)6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−0.186 + 2.99i)9-s + (−0.686 − 2.12i)10-s + (0.813 − 0.469i)11-s + (−0.500 + 1.65i)12-s − 2·13-s + (−2 + 1.73i)14-s + (−3.87 + 0.0737i)15-s + (−0.5 + 0.866i)16-s + (5.74 − 3.31i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.684 − 0.728i)3-s + (−0.249 − 0.433i)4-s + (0.670 − 0.741i)5-s + (−0.688 + 0.161i)6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−0.0620 + 0.998i)9-s + (−0.216 − 0.672i)10-s + (0.245 − 0.141i)11-s + (−0.144 + 0.478i)12-s − 0.554·13-s + (−0.534 + 0.462i)14-s + (−0.999 + 0.0190i)15-s + (−0.125 + 0.216i)16-s + (1.39 − 0.804i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.319579 - 1.02221i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.319579 - 1.02221i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.18 + 1.26i)T \) |
| 5 | \( 1 + (-1.5 + 1.65i)T \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 11 | \( 1 + (-0.813 + 0.469i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-5.74 + 3.31i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.686 - 1.18i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.31iT - 29T^{2} \) |
| 31 | \( 1 + (-6.55 + 3.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.11 - 4.10i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 1.08iT - 43T^{2} \) |
| 47 | \( 1 + (-7.37 - 4.25i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.18 + 3.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.55 + 11.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (11.0 + 6.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.05 - 1.18i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 + (0.686 - 1.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−12.29710024577713072487104038690, −11.20029487959412788916310963184, −10.03750497878771617823655569701, −9.394499244420321066657714326897, −7.86508457166676044497438649880, −6.53848819121091112833528499951, −5.67408968188284297306057447374, −4.55532473929117568496186694290, −2.68727074443064898284934803942, −0.937759310051651592901227003732,
2.99766768252635341737852230372, 4.28538566094244230463954409208, 5.88868914188935807420632921639, 6.12444944575195339898110914673, 7.41956053095984891250775878257, 9.031961126467121765048246567950, 9.956130264270265554536980611346, 10.55639371946922052186029091092, 12.02887913963393400268271948799, 12.61795935744981002616865455632