L(s) = 1 | − 3.79i·2-s − 6.38·4-s + 19.9·5-s + (17.7 + 5.44i)7-s − 6.12i·8-s − 75.6i·10-s + 35.9i·11-s − 7.62i·13-s + (20.6 − 67.1i)14-s − 74.3·16-s − 21.4·17-s − 97.3i·19-s − 127.·20-s + 136.·22-s + 41.2i·23-s + ⋯ |
L(s) = 1 | − 1.34i·2-s − 0.798·4-s + 1.78·5-s + (0.955 + 0.294i)7-s − 0.270i·8-s − 2.39i·10-s + 0.986i·11-s − 0.162i·13-s + (0.394 − 1.28i)14-s − 1.16·16-s − 0.306·17-s − 1.17i·19-s − 1.42·20-s + 1.32·22-s + 0.373i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.233631885\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.233631885\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-17.7 - 5.44i)T \) |
good | 2 | \( 1 + 3.79iT - 8T^{2} \) |
| 5 | \( 1 - 19.9T + 125T^{2} \) |
| 11 | \( 1 - 35.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 7.62iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 21.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 41.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 59.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 70.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 355.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 14.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 469.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 710. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 465.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 626. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 524.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 115. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 708. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 401.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 977.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 601. iT - 9.12e5T^{2} \) |
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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
−10.11472158076844093463051574392, −9.502120938875891815685671792684, −8.846968722475286130645181739474, −7.36352594198710432756833015004, −6.28285934042162541958909960133, −5.20313770229528461440836942872, −4.34505597077638180700954292585, −2.58427864266426937605596849215, −2.12413669696288646710927309140, −1.09888150170969001872133117297,
1.34301989156356958272180952500, 2.52693660962779531855588063824, 4.43730579065257576261062374865, 5.54614550275537415565011256576, 5.93060121913175117044717243210, 6.82755833996891576955034461643, 7.900862255358345030755431498063, 8.667762121745616590995874714011, 9.451483101833082722482249712298, 10.56100476663684217365126325198