| L(s) = 1 | + 37.7·2-s + 186.·3-s + 915.·4-s − 1.74e3·5-s + 7.03e3·6-s − 6.90e3·7-s + 1.52e4·8-s + 1.50e4·9-s − 6.60e4·10-s + 1.46e4·11-s + 1.70e5·12-s + 1.76e5·13-s − 2.60e5·14-s − 3.25e5·15-s + 1.07e5·16-s − 4.78e5·17-s + 5.67e5·18-s + 1.93e5·19-s − 1.59e6·20-s − 1.28e6·21-s + 5.53e5·22-s + 1.49e6·23-s + 2.84e6·24-s + 1.09e6·25-s + 6.68e6·26-s − 8.69e5·27-s − 6.32e6·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.32·3-s + 1.78·4-s − 1.24·5-s + 2.21·6-s − 1.08·7-s + 1.31·8-s + 0.762·9-s − 2.08·10-s + 0.301·11-s + 2.37·12-s + 1.71·13-s − 1.81·14-s − 1.65·15-s + 0.410·16-s − 1.38·17-s + 1.27·18-s + 0.340·19-s − 2.23·20-s − 1.44·21-s + 0.503·22-s + 1.11·23-s + 1.74·24-s + 0.562·25-s + 2.86·26-s − 0.314·27-s − 1.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.200483162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.200483162\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 1.46e4T \) |
| good | 2 | \( 1 - 37.7T + 512T^{2} \) |
| 3 | \( 1 - 186.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.74e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 6.90e3T + 4.03e7T^{2} \) |
| 13 | \( 1 - 1.76e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.78e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.93e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.49e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 9.21e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.10e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.90e5T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.56e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.40e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.12e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.17e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.97e6T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.92e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.43e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.83e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.96e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.88e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.19e8T + 7.60e17T^{2} \) |
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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
−19.12829485506666934690130420583, −15.80185872499729199867511332597, −15.40182877103290554644273789145, −13.85045899358561158995425146420, −12.99121280975151603865349800839, −11.40019851822326925450470701114, −8.669157828527143466450931697778, −6.69630746961796034834340348926, −3.99481267042636135312324427559, −3.08208623811248569873831264729,
3.08208623811248569873831264729, 3.99481267042636135312324427559, 6.69630746961796034834340348926, 8.669157828527143466450931697778, 11.40019851822326925450470701114, 12.99121280975151603865349800839, 13.85045899358561158995425146420, 15.40182877103290554644273789145, 15.80185872499729199867511332597, 19.12829485506666934690130420583
