| L(s) = 1 | − 25.4·2-s + 80.7·3-s + 135.·4-s − 783.·5-s − 2.05e3·6-s − 5.74e3·7-s + 9.57e3·8-s − 1.31e4·9-s + 1.99e4·10-s + 2.59e3·11-s + 1.09e4·12-s − 1.08e5·13-s + 1.46e5·14-s − 6.32e4·15-s − 3.13e5·16-s + 3.34e5·18-s + 9.56e4·19-s − 1.06e5·20-s − 4.64e5·21-s − 6.59e4·22-s − 9.29e4·23-s + 7.73e5·24-s − 1.33e6·25-s + 2.77e6·26-s − 2.65e6·27-s − 7.79e5·28-s − 2.91e6·29-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.575·3-s + 0.265·4-s − 0.560·5-s − 0.647·6-s − 0.904·7-s + 0.826·8-s − 0.668·9-s + 0.630·10-s + 0.0534·11-s + 0.152·12-s − 1.05·13-s + 1.01·14-s − 0.322·15-s − 1.19·16-s + 0.751·18-s + 0.168·19-s − 0.148·20-s − 0.520·21-s − 0.0600·22-s − 0.0692·23-s + 0.475·24-s − 0.685·25-s + 1.18·26-s − 0.960·27-s − 0.239·28-s − 0.765·29-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
| \(L(5)\) |
\(\approx\) |
\(8.174174931\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.174174931\times10^{-5}\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 25.4T + 512T^{2} \) |
| 3 | \( 1 - 80.7T + 1.96e4T^{2} \) |
| 5 | \( 1 + 783.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.74e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.59e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.08e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 9.56e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 9.29e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.91e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.18e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.09e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.11e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.20e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.88e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.85e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.51e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.53e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.05e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.16e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.90e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 9.72e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.27e9T + 7.60e17T^{2} \) |
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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
−9.783823899401154046449273478511, −9.379730410316379003656166630310, −8.369496721400139806418803388275, −7.68831151118326562768024336931, −6.77471038110801648805339834149, −5.31187619662712107657981461942, −3.94991906309660907607095396801, −2.92257361234466736215413861152, −1.72918201498605782931087561798, −0.00447442895901496725607574186,
0.00447442895901496725607574186, 1.72918201498605782931087561798, 2.92257361234466736215413861152, 3.94991906309660907607095396801, 5.31187619662712107657981461942, 6.77471038110801648805339834149, 7.68831151118326562768024336931, 8.369496721400139806418803388275, 9.379730410316379003656166630310, 9.783823899401154046449273478511
