| L(s) = 1 | + 2.64·2-s + 62.7·3-s − 504.·4-s + 1.95e3·5-s + 166.·6-s + 4.15e3·7-s − 2.69e3·8-s − 1.57e4·9-s + 5.18e3·10-s + 7.30e4·11-s − 3.16e4·12-s − 5.43e4·13-s + 1.09e4·14-s + 1.22e5·15-s + 2.51e5·16-s − 4.17e4·18-s + 8.11e5·19-s − 9.88e5·20-s + 2.60e5·21-s + 1.93e5·22-s + 1.63e5·23-s − 1.68e5·24-s + 1.87e6·25-s − 1.43e5·26-s − 2.22e6·27-s − 2.09e6·28-s − 5.42e6·29-s + ⋯ |
| L(s) = 1 | + 0.117·2-s + 0.446·3-s − 0.986·4-s + 1.40·5-s + 0.0523·6-s + 0.653·7-s − 0.232·8-s − 0.800·9-s + 0.163·10-s + 1.50·11-s − 0.440·12-s − 0.527·13-s + 0.0764·14-s + 0.626·15-s + 0.959·16-s − 0.0936·18-s + 1.42·19-s − 1.38·20-s + 0.292·21-s + 0.176·22-s + 0.121·23-s − 0.103·24-s + 0.962·25-s − 0.0617·26-s − 0.804·27-s − 0.644·28-s − 1.42·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.510677766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.510677766\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 512T^{2} \) |
| 3 | \( 1 - 62.7T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.95e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.15e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.30e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 5.43e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.11e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.63e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.46e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.91e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.92e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.75e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.01e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.10e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.39e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.92e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.76e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.85e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.16e8T + 7.60e17T^{2} \) |
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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
−9.688514216470803372564932569041, −9.433779717364747903852312855019, −8.615650958739302753964140971321, −7.46952792883232599896399862011, −5.99405035469675253711371954693, −5.39198278140567022828266485797, −4.24396991782033893395212919689, −3.05028843721538231402298713363, −1.82946156065345199627482421888, −0.849376933474067797143706225051,
0.849376933474067797143706225051, 1.82946156065345199627482421888, 3.05028843721538231402298713363, 4.24396991782033893395212919689, 5.39198278140567022828266485797, 5.99405035469675253711371954693, 7.46952792883232599896399862011, 8.615650958739302753964140971321, 9.433779717364747903852312855019, 9.688514216470803372564932569041
