| L(s) = 1 | + 43.4·2-s + 157.·3-s + 1.37e3·4-s + 2.50e3·5-s + 6.82e3·6-s − 5.91e3·7-s + 3.75e4·8-s + 4.98e3·9-s + 1.08e5·10-s + 6.77e3·11-s + 2.16e5·12-s − 1.23e4·13-s − 2.57e5·14-s + 3.93e5·15-s + 9.25e5·16-s + 2.16e5·18-s + 9.46e4·19-s + 3.44e6·20-s − 9.29e5·21-s + 2.94e5·22-s − 1.61e6·23-s + 5.89e6·24-s + 4.32e6·25-s − 5.34e5·26-s − 2.30e6·27-s − 8.13e6·28-s − 4.52e5·29-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 1.11·3-s + 2.68·4-s + 1.79·5-s + 2.14·6-s − 0.931·7-s + 3.23·8-s + 0.253·9-s + 3.44·10-s + 0.139·11-s + 3.00·12-s − 0.119·13-s − 1.78·14-s + 2.00·15-s + 3.53·16-s + 0.486·18-s + 0.166·19-s + 4.81·20-s − 1.04·21-s + 0.267·22-s − 1.20·23-s + 3.62·24-s + 2.21·25-s − 0.229·26-s − 0.836·27-s − 2.50·28-s − 0.118·29-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
| \(L(5)\) |
\(\approx\) |
\(16.57065870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.57065870\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 43.4T + 512T^{2} \) |
| 3 | \( 1 - 157.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.50e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.91e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.77e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.23e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 9.46e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.61e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.52e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.43e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.76e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.78e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.47e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.18e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.64e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.04e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.99e5T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.92e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.45e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.99e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.92e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.69e9T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08137761980930484568403830407, −9.632052218695557651454291727191, −8.170917836499743293414763256325, −6.74524038288747009316620426091, −6.15499268547909346133119950536, −5.33600081916876662107675817093, −4.07309554492963291721858669064, −2.94340990417596873527994155426, −2.49852372298866054298563846806, −1.54472452943415834639093416844,
1.54472452943415834639093416844, 2.49852372298866054298563846806, 2.94340990417596873527994155426, 4.07309554492963291721858669064, 5.33600081916876662107675817093, 6.15499268547909346133119950536, 6.74524038288747009316620426091, 8.170917836499743293414763256325, 9.632052218695557651454291727191, 10.08137761980930484568403830407
