L(s) = 1 | − 2-s − 3.23·3-s + 4-s − 2.83·5-s + 3.23·6-s − 0.371·7-s − 8-s + 7.48·9-s + 2.83·10-s + 1.92·11-s − 3.23·12-s + 6.61·13-s + 0.371·14-s + 9.19·15-s + 16-s − 17-s − 7.48·18-s + 1.69·19-s − 2.83·20-s + 1.20·21-s − 1.92·22-s − 4.71·23-s + 3.23·24-s + 3.06·25-s − 6.61·26-s − 14.5·27-s − 0.371·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.86·3-s + 0.5·4-s − 1.26·5-s + 1.32·6-s − 0.140·7-s − 0.353·8-s + 2.49·9-s + 0.897·10-s + 0.581·11-s − 0.934·12-s + 1.83·13-s + 0.0993·14-s + 2.37·15-s + 0.250·16-s − 0.242·17-s − 1.76·18-s + 0.388·19-s − 0.634·20-s + 0.262·21-s − 0.411·22-s − 0.983·23-s + 0.660·24-s + 0.612·25-s − 1.29·26-s − 2.79·27-s − 0.0702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4043487372\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4043487372\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 + 0.371T + 7T^{2} \) |
| 11 | \( 1 - 1.92T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 + 4.71T + 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 - 8.05T + 53T^{2} \) |
| 61 | \( 1 - 9.01T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 + 5.91T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 3.65T + 97T^{2} \) |
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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
−9.199245045099084764982447265668, −8.300949609732462933299858605862, −7.47216973733994384302988970365, −6.78175358530443705991703484248, −6.08205103818559417020838396983, −5.40609002304197513719978252381, −4.09087922149301993282581304203, −3.72382474543052616457682905821, −1.56901037521958480071663946420, −0.54821248269424041287164770055,
0.54821248269424041287164770055, 1.56901037521958480071663946420, 3.72382474543052616457682905821, 4.09087922149301993282581304203, 5.40609002304197513719978252381, 6.08205103818559417020838396983, 6.78175358530443705991703484248, 7.47216973733994384302988970365, 8.300949609732462933299858605862, 9.199245045099084764982447265668