L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 7-s − 0.999·8-s + (1.5 − 2.59i)9-s + 5·11-s + (1 − 1.73i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 3·18-s + (−0.5 − 4.33i)19-s + (2.5 + 4.33i)22-s + (0.5 − 0.866i)23-s + 1.99·26-s + (0.499 − 0.866i)28-s + (−3 + 5.19i)29-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.377·7-s − 0.353·8-s + (0.5 − 0.866i)9-s + 1.50·11-s + (0.277 − 0.480i)13-s + (−0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.707·18-s + (−0.114 − 0.993i)19-s + (0.533 + 0.923i)22-s + (0.104 − 0.180i)23-s + 0.392·26-s + (0.0944 − 0.163i)28-s + (−0.557 + 0.964i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94884 + 0.421198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94884 + 0.421198i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.5 + 4.33i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 + 4.33i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.719832933752611909378027284309, −9.317519993778239477508888710175, −8.433991680114459480987162768451, −7.35155301191210300128862824800, −6.51779602412075469500115003179, −6.11675240902167177274167002985, −4.73339480470200908725662950753, −3.92842817128213055915857811861, −2.98054166850019483021196192632, −1.04987745002552233066082512272,
1.29793947836325581910441911715, 2.40931606491552331823877825361, 3.87614472238841220322712760618, 4.29353267796000027187381777841, 5.64848032313846357543304691273, 6.41831436480062518115554336333, 7.40655059846664527146659403522, 8.447918354250738830884132284406, 9.430729454746527553327742374237, 9.932918427995837414089519391319