L(s) = 1 | + (−99.2 + 99.2i)3-s + (1.73e4 + 1.73e4i)7-s − 1.96e4i·9-s − 2.50e4·11-s + (−2.00e5 + 2.00e5i)13-s + (1.64e6 + 1.64e6i)17-s + 1.78e6i·19-s − 3.43e6·21-s + (2.10e6 − 2.10e6i)23-s + (1.95e6 + 1.95e6i)27-s − 4.57e5i·29-s − 1.63e7·31-s + (2.48e6 − 2.48e6i)33-s + (3.60e7 + 3.60e7i)37-s − 3.98e7i·39-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (1.03 + 1.03i)7-s − 0.333i·9-s − 0.155·11-s + (−0.540 + 0.540i)13-s + (1.15 + 1.15i)17-s + 0.719i·19-s − 0.841·21-s + (0.327 − 0.327i)23-s + (0.136 + 0.136i)27-s − 0.0223i·29-s − 0.569·31-s + (0.0634 − 0.0634i)33-s + (0.519 + 0.519i)37-s − 0.441i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.776191594\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776191594\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (99.2 - 99.2i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-1.73e4 - 1.73e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + 2.50e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + (2.00e5 - 2.00e5i)T - 1.37e11iT^{2} \) |
| 17 | \( 1 + (-1.64e6 - 1.64e6i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 - 1.78e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (-2.10e6 + 2.10e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 + 4.57e5iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 1.63e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (-3.60e7 - 3.60e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + 1.20e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + (1.74e8 - 1.74e8i)T - 2.16e16iT^{2} \) |
| 47 | \( 1 + (-2.51e8 - 2.51e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 + (-1.67e8 + 1.67e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 - 9.73e5iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 7.13e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + (3.85e8 + 3.85e8i)T + 1.82e18iT^{2} \) |
| 71 | \( 1 - 2.86e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.05e9 + 2.05e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + 1.26e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (1.02e9 - 1.02e9i)T - 1.55e19iT^{2} \) |
| 89 | \( 1 - 4.06e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (1.33e9 + 1.33e9i)T + 7.37e19iT^{2} \) |
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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
−10.43446498307240545859821214210, −9.562310253178935364847351054976, −8.499409261173675386903919825543, −7.78886443098712012229810430671, −6.37199134045208175799686180780, −5.46154450083789185662753361900, −4.73242530581545176068652869177, −3.52695488746028923543718967818, −2.18284055351329107015592592547, −1.24173785114494554175147296270,
0.38831492963380222619969679053, 1.05252320846996518449768010455, 2.27391834606179108544622752445, 3.59267984676088997062831329160, 4.90941181798241637998737210382, 5.43577891198701288866187492395, 7.07121966386355792595180045548, 7.43298586223730647084567056893, 8.444741030648171431133401556737, 9.779568155622515695578748438329