| L(s) = 1 | + 9.92·2-s + 0.133·3-s + 66.4·4-s + 25·5-s + 1.32·6-s − 49·7-s + 342.·8-s − 242.·9-s + 248.·10-s + 674.·11-s + 8.87·12-s − 770.·13-s − 486.·14-s + 3.33·15-s + 1.26e3·16-s − 693.·17-s − 2.41e3·18-s − 1.39e3·19-s + 1.66e3·20-s − 6.53·21-s + 6.69e3·22-s + 1.32e3·23-s + 45.6·24-s + 625·25-s − 7.64e3·26-s − 64.8·27-s − 3.25e3·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.00855·3-s + 2.07·4-s + 0.447·5-s + 0.0150·6-s − 0.377·7-s + 1.89·8-s − 0.999·9-s + 0.784·10-s + 1.68·11-s + 0.0177·12-s − 1.26·13-s − 0.663·14-s + 0.00382·15-s + 1.23·16-s − 0.582·17-s − 1.75·18-s − 0.884·19-s + 0.929·20-s − 0.00323·21-s + 2.94·22-s + 0.521·23-s + 0.0161·24-s + 0.200·25-s − 2.21·26-s − 0.0171·27-s − 0.785·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.885128296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.885128296\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
| good | 2 | \( 1 - 9.92T + 32T^{2} \) |
| 3 | \( 1 - 0.133T + 243T^{2} \) |
| 11 | \( 1 - 674.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 770.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 693.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 172.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.07e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.54e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.72e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.90e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.27e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.01e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.08e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.66e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5T + 8.58e9T^{2} \) |
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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
−14.78288737876691218861002040676, −14.48243282232056185854499427606, −13.19343360617685619838370965677, −12.15296192092692054900487640883, −11.13652586726635921134251282059, −9.224983301120144057037065089776, −6.85665198503451830697754763124, −5.79364497774359238453080092302, −4.22354078797919766698321345583, −2.53819572753356240530969027156,
2.53819572753356240530969027156, 4.22354078797919766698321345583, 5.79364497774359238453080092302, 6.85665198503451830697754763124, 9.224983301120144057037065089776, 11.13652586726635921134251282059, 12.15296192092692054900487640883, 13.19343360617685619838370965677, 14.48243282232056185854499427606, 14.78288737876691218861002040676
