L(s) = 1 | + (2.86 − 15.3i)3-s − 32·4-s + 49.9i·5-s − 258.·7-s + (−226. − 87.6i)9-s + (−91.5 + 490. i)12-s + (764. + 142. i)15-s + 1.02e3·16-s − 683. i·17-s + 896.·19-s − 1.59e3i·20-s + (−739. + 3.96e3i)21-s + 633.·25-s + (−1.99e3 + 3.22e3i)27-s + 8.27e3·28-s + 3.24e3i·29-s + ⋯ |
L(s) = 1 | + (0.183 − 0.983i)3-s − 4-s + 0.892i·5-s − 1.99·7-s + (−0.932 − 0.360i)9-s + (−0.183 + 0.983i)12-s + (0.877 + 0.163i)15-s + 16-s − 0.573i·17-s + 0.569·19-s − 0.892i·20-s + (−0.366 + 1.96i)21-s + 0.202·25-s + (−0.525 + 0.850i)27-s + 1.99·28-s + 0.716i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8339405406\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8339405406\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.86 + 15.3i)T \) |
| 59 | \( 1 + 2.67e4iT \) |
good | 2 | \( 1 + 32T^{2} \) |
| 5 | \( 1 - 49.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 258.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 + 683. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 896.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.24e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.89e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.88e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.94e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 - 9.68e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−12.17091503078063237256315904961, −10.64309598584144918699037118574, −9.594245969108183982232889377730, −8.873348688376130404954503814241, −7.40182431523670109167041251588, −6.65289216354584844539371430711, −5.60940791421029063036682178827, −3.58366243921556658688691226213, −2.79199418591818840342852145919, −0.61780575586255061420920796622,
0.50366992135817344513325016886, 3.13955587279269194280119004764, 4.05747064257536914831777519991, 5.17662213856628252579482979317, 6.25838545534191833109995519271, 8.129824788387795597931209572356, 9.144465433626853279034278621073, 9.604486743201143930796677914141, 10.38524875733519203901941472901, 12.03192249500921121902691688446