L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (4.47 − 2.58i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + 0.999i·12-s + (1.87 − 3.08i)13-s + 5.16·14-s + (−0.5 + 0.866i)16-s + (4.47 − 2.58i)17-s + 0.999i·18-s + (−0.581 − 1.00i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (1.68 − 0.975i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.150 + 0.261i)11-s + 0.288i·12-s + (0.519 − 0.854i)13-s + 1.37·14-s + (−0.125 + 0.216i)16-s + (1.08 − 0.626i)17-s + 0.235i·18-s + (−0.133 − 0.230i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.735836168\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.735836168\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.87 + 3.08i)T \) |
good | 7 | \( 1 + (-4.47 + 2.58i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.47 + 2.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.581 + 1.00i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.60 + 2.08i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.16 + 3.74i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.16T + 31T^{2} \) |
| 37 | \( 1 + (5.33 + 3.08i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.58 - 4.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.73 + 1.58i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.08 + 5.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.24 - 7.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.32iT - 73T^{2} \) |
| 79 | \( 1 + 5.48T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 + (-3.74 + 6.48i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.34 - 3.66i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.023637179733030467047912463920, −8.117961295479890953511618788199, −7.74315733170156254721619885214, −7.10719559810405457427001530664, −5.79207192011194611199822738217, −5.08397165813977897509252861226, −4.32717573693437045227334411759, −3.61466158501532225536387648791, −2.44201183037425136157062243092, −1.20592681270381483151500825509,
1.60124901110636547712790806918, 1.95123151662546194562566207789, 3.29457328807210026436707551534, 4.11168220590802833849034717622, 5.21026461791967083270046926959, 5.66070840354109154687518153259, 6.72974262225744244932363446016, 7.75005671669057468269166389624, 8.398496889056116080842532338567, 8.937326825706588730118711324430