L(s) = 1 | − 2.11·2-s + 3-s + 2.48·4-s + 0.996·5-s − 2.11·6-s − 4.64·7-s − 1.03·8-s + 9-s − 2.11·10-s − 2.11·11-s + 2.48·12-s − 4.22·13-s + 9.84·14-s + 0.996·15-s − 2.79·16-s − 0.208·17-s − 2.11·18-s + 0.542·19-s + 2.47·20-s − 4.64·21-s + 4.48·22-s + 5.01·23-s − 1.03·24-s − 4.00·25-s + 8.95·26-s + 27-s − 11.5·28-s + ⋯ |
L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.24·4-s + 0.445·5-s − 0.864·6-s − 1.75·7-s − 0.364·8-s + 0.333·9-s − 0.667·10-s − 0.638·11-s + 0.717·12-s − 1.17·13-s + 2.63·14-s + 0.257·15-s − 0.697·16-s − 0.0506·17-s − 0.499·18-s + 0.124·19-s + 0.553·20-s − 1.01·21-s + 0.956·22-s + 1.04·23-s − 0.210·24-s − 0.801·25-s + 1.75·26-s + 0.192·27-s − 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4925767683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4925767683\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 5 | \( 1 - 0.996T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 + 2.11T + 11T^{2} \) |
| 13 | \( 1 + 4.22T + 13T^{2} \) |
| 17 | \( 1 + 0.208T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 23 | \( 1 - 5.01T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 - 8.67T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 2.13T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 6.98T + 61T^{2} \) |
| 67 | \( 1 + 0.434T + 67T^{2} \) |
| 71 | \( 1 + 8.13T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 + 3.11T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
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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
−8.093305826348624349516237305481, −7.50977889630834740775651919834, −6.94243841700389792619991689541, −6.30515502595392292362392440256, −5.37250015250945059188777332441, −4.35739520294338618314178203695, −3.16141470776769280106762317087, −2.66474173183508144110799602689, −1.77224960974228289945236414948, −0.42957847305157577252838157275,
0.42957847305157577252838157275, 1.77224960974228289945236414948, 2.66474173183508144110799602689, 3.16141470776769280106762317087, 4.35739520294338618314178203695, 5.37250015250945059188777332441, 6.30515502595392292362392440256, 6.94243841700389792619991689541, 7.50977889630834740775651919834, 8.093305826348624349516237305481