L(s) = 1 | + (−10.3 − 11.6i)3-s + 68.3i·5-s + 220. i·7-s + (−28.4 + 241. i)9-s + 127.·11-s + 822.·13-s + (796. − 707. i)15-s + 1.61e3i·17-s + 1.81e3i·19-s + (2.57e3 − 2.28e3i)21-s + 1.27e3·23-s − 1.54e3·25-s + (3.10e3 − 2.16e3i)27-s + 7.51e3i·29-s − 8.58e3i·31-s + ⋯ |
L(s) = 1 | + (−0.664 − 0.747i)3-s + 1.22i·5-s + 1.70i·7-s + (−0.117 + 0.993i)9-s + 0.316·11-s + 1.34·13-s + (0.913 − 0.812i)15-s + 1.35i·17-s + 1.15i·19-s + (1.27 − 1.13i)21-s + 0.501·23-s − 0.493·25-s + (0.820 − 0.572i)27-s + 1.65i·29-s − 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.688685486\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688685486\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (10.3 + 11.6i)T \) |
good | 5 | \( 1 - 68.3iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 220. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 127.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 822.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.61e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 8.58e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 5.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.07e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 4.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.95e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.46e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.40e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.95e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.97e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−11.06343528193114551045977215103, −10.20532264308872830441631858021, −8.800821268763435187084804123673, −8.092586717391249754392642582964, −6.82077407007536447494630789631, −6.06890973278475962982093594374, −5.56966872617726636274833161838, −3.69543215663574202068029206663, −2.46656009373489751919152882789, −1.45986035802684234415533365337,
0.60282495702453514548557770536, 0.994306549891834787683896177866, 3.43201331540289916582247267257, 4.44264415326452166219332635211, 4.95974333193765876176319743892, 6.31126558849837349679022929833, 7.24717044057913706511460766644, 8.566745780147136207931870966808, 9.343051988820995529722670566764, 10.20268250454914981452231688590