| L(s) = 1 | − 10.3·2-s + 20.6·3-s + 76.0·4-s + 8.64·5-s − 214.·6-s + 164.·7-s − 458.·8-s + 183.·9-s − 89.9·10-s + 121·11-s + 1.57e3·12-s − 585.·13-s − 1.70e3·14-s + 178.·15-s + 2.33e3·16-s − 945.·17-s − 1.90e3·18-s + 1.14e3·19-s + 658.·20-s + 3.39e3·21-s − 1.25e3·22-s − 1.34e3·23-s − 9.46e3·24-s − 3.05e3·25-s + 6.08e3·26-s − 1.23e3·27-s + 1.25e4·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 1.32·3-s + 2.37·4-s + 0.154·5-s − 2.43·6-s + 1.26·7-s − 2.53·8-s + 0.754·9-s − 0.284·10-s + 0.301·11-s + 3.14·12-s − 0.960·13-s − 2.33·14-s + 0.204·15-s + 2.27·16-s − 0.793·17-s − 1.38·18-s + 0.730·19-s + 0.367·20-s + 1.68·21-s − 0.554·22-s − 0.530·23-s − 3.35·24-s − 0.976·25-s + 1.76·26-s − 0.325·27-s + 3.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8699849810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8699849810\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 3 | \( 1 - 20.6T + 243T^{2} \) |
| 5 | \( 1 - 8.64T + 3.12e3T^{2} \) |
| 7 | \( 1 - 164.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 585.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 945.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.14e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 899.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 390.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.60e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.99e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.53e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 325.T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.01e3T + 8.58e9T^{2} \) |
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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
−19.59836807842209357915381307988, −18.19724209719430998528422779481, −17.22343104291337690704786546439, −15.46760807784459608929348040494, −14.23740802217302320889279843020, −11.52665372202715867775674536269, −9.787141937630919361056743990519, −8.565526045186280179248112159778, −7.52056970875392553016543902856, −2.06717744522228783734671585055,
2.06717744522228783734671585055, 7.52056970875392553016543902856, 8.565526045186280179248112159778, 9.787141937630919361056743990519, 11.52665372202715867775674536269, 14.23740802217302320889279843020, 15.46760807784459608929348040494, 17.22343104291337690704786546439, 18.19724209719430998528422779481, 19.59836807842209357915381307988
