| L(s) = 1 | − 10.6·2-s + 80.8·4-s − 67.4·5-s − 518.·8-s + 716.·10-s + 522.·11-s − 76.6·13-s + 2.92e3·16-s − 1.26e3·17-s + 1.89e3·19-s − 5.45e3·20-s − 5.55e3·22-s − 1.15e3·23-s + 1.42e3·25-s + 814.·26-s − 3.85e3·29-s + 1.04e4·31-s − 1.44e4·32-s + 1.34e4·34-s − 5.60e3·37-s − 2.01e4·38-s + 3.50e4·40-s + 1.42e4·41-s − 1.48e4·43-s + 4.22e4·44-s + 1.22e4·46-s + 1.15e4·47-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.52·4-s − 1.20·5-s − 2.86·8-s + 2.26·10-s + 1.30·11-s − 0.125·13-s + 2.85·16-s − 1.06·17-s + 1.20·19-s − 3.04·20-s − 2.44·22-s − 0.453·23-s + 0.456·25-s + 0.236·26-s − 0.850·29-s + 1.94·31-s − 2.49·32-s + 2.00·34-s − 0.672·37-s − 2.25·38-s + 3.45·40-s + 1.32·41-s − 1.22·43-s + 3.28·44-s + 0.851·46-s + 0.762·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5224717885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5224717885\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 5 | \( 1 + 67.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 522.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 76.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.26e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.04e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.67e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.60e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.83e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.61e4T + 8.58e9T^{2} \) |
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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
−10.11541960927741364985313367253, −9.268633145897075923846927575658, −8.591208651528759991101900264192, −7.72185494617530194179263472509, −7.06052952313321837268139917208, −6.15504681988190500826502723601, −4.26999520672535621404230216818, −3.02711212529684676516252857779, −1.58684139249484044742526990129, −0.51288441384034722742801495460,
0.51288441384034722742801495460, 1.58684139249484044742526990129, 3.02711212529684676516252857779, 4.26999520672535621404230216818, 6.15504681988190500826502723601, 7.06052952313321837268139917208, 7.72185494617530194179263472509, 8.591208651528759991101900264192, 9.268633145897075923846927575658, 10.11541960927741364985313367253
