| L(s) = 1 | − 17.1·2-s + 244.·3-s − 217.·4-s + 2.66e3·5-s − 4.18e3·6-s + 2.40e3·7-s + 1.25e4·8-s + 3.99e4·9-s − 4.57e4·10-s + 5.28e4·11-s − 5.31e4·12-s − 2.85e4·13-s − 4.11e4·14-s + 6.50e5·15-s − 1.03e5·16-s + 2.47e5·17-s − 6.84e5·18-s + 6.91e4·19-s − 5.80e5·20-s + 5.86e5·21-s − 9.05e5·22-s − 9.89e4·23-s + 3.05e6·24-s + 5.15e6·25-s + 4.90e5·26-s + 4.93e6·27-s − 5.22e5·28-s + ⋯ |
| L(s) = 1 | − 0.758·2-s + 1.73·3-s − 0.425·4-s + 1.90·5-s − 1.31·6-s + 0.377·7-s + 1.08·8-s + 2.02·9-s − 1.44·10-s + 1.08·11-s − 0.739·12-s − 0.277·13-s − 0.286·14-s + 3.31·15-s − 0.394·16-s + 0.718·17-s − 1.53·18-s + 0.121·19-s − 0.810·20-s + 0.657·21-s − 0.824·22-s − 0.0737·23-s + 1.88·24-s + 2.63·25-s + 0.210·26-s + 1.78·27-s − 0.160·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.825926920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.825926920\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 2 | \( 1 + 17.1T + 512T^{2} \) |
| 3 | \( 1 - 244.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.66e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 5.28e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 2.47e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.91e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.89e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.43e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.61e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.17e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.50e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.08e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.74e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.47e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.75e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.18e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.31e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.82e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.89e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.06e9T + 7.60e17T^{2} \) |
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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
−12.72449410307600663469205380674, −10.48950826398621377379479136884, −9.470778968882247028443458035587, −9.264561527609059686185035148584, −8.218599823267884635106993221544, −6.94958265966285259292653817919, −5.18554522428844654366906249157, −3.56265602408258733953596853813, −1.96854342225959871955601401414, −1.42182065475238225539366024584,
1.42182065475238225539366024584, 1.96854342225959871955601401414, 3.56265602408258733953596853813, 5.18554522428844654366906249157, 6.94958265966285259292653817919, 8.218599823267884635106993221544, 9.264561527609059686185035148584, 9.470778968882247028443458035587, 10.48950826398621377379479136884, 12.72449410307600663469205380674
