L(s) = 1 | + (−40.5 + 80.9i)2-s + (−1.63e3 + 1.63e3i)3-s + (−4.90e3 − 6.55e3i)4-s + (9.55e3 + 9.55e3i)5-s + (−6.62e4 − 1.99e5i)6-s − 1.33e5i·7-s + (7.29e5 − 1.31e5i)8-s − 3.78e6i·9-s + (−1.16e6 + 3.86e5i)10-s + (−3.83e5 − 3.83e5i)11-s + (1.88e7 + 2.70e6i)12-s + (−1.38e6 + 1.38e6i)13-s + (1.08e7 + 5.42e6i)14-s − 3.13e7·15-s + (−1.89e7 + 6.43e7i)16-s − 1.51e8·17-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.894i)2-s + (−1.29 + 1.29i)3-s + (−0.599 − 0.800i)4-s + (0.273 + 0.273i)5-s + (−0.579 − 1.74i)6-s − 0.430i·7-s + (0.984 − 0.177i)8-s − 2.37i·9-s + (−0.367 + 0.122i)10-s + (−0.0651 − 0.0651i)11-s + (1.81 + 0.261i)12-s + (−0.0794 + 0.0794i)13-s + (0.384 + 0.192i)14-s − 0.710·15-s + (−0.282 + 0.959i)16-s − 1.52·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(0.579969 + 0.277324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.579969 + 0.277324i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (40.5 - 80.9i)T \) |
good | 3 | \( 1 + (1.63e3 - 1.63e3i)T - 1.59e6iT^{2} \) |
| 5 | \( 1 + (-9.55e3 - 9.55e3i)T + 1.22e9iT^{2} \) |
| 7 | \( 1 + 1.33e5iT - 9.68e10T^{2} \) |
| 11 | \( 1 + (3.83e5 + 3.83e5i)T + 3.45e13iT^{2} \) |
| 13 | \( 1 + (1.38e6 - 1.38e6i)T - 3.02e14iT^{2} \) |
| 17 | \( 1 + 1.51e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + (-1.91e8 + 1.91e8i)T - 4.20e16iT^{2} \) |
| 23 | \( 1 - 1.33e9iT - 5.04e17T^{2} \) |
| 29 | \( 1 + (-2.13e9 + 2.13e9i)T - 1.02e19iT^{2} \) |
| 31 | \( 1 - 3.46e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (2.02e10 + 2.02e10i)T + 2.43e20iT^{2} \) |
| 41 | \( 1 + 5.32e9iT - 9.25e20T^{2} \) |
| 43 | \( 1 + (-2.42e10 - 2.42e10i)T + 1.71e21iT^{2} \) |
| 47 | \( 1 + 4.54e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + (-1.29e11 - 1.29e11i)T + 2.60e22iT^{2} \) |
| 59 | \( 1 + (-2.31e11 - 2.31e11i)T + 1.04e23iT^{2} \) |
| 61 | \( 1 + (-3.36e11 + 3.36e11i)T - 1.61e23iT^{2} \) |
| 67 | \( 1 + (-1.18e11 + 1.18e11i)T - 5.48e23iT^{2} \) |
| 71 | \( 1 + 8.45e10iT - 1.16e24T^{2} \) |
| 73 | \( 1 - 1.06e12iT - 1.67e24T^{2} \) |
| 79 | \( 1 - 6.11e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + (-1.46e12 + 1.46e12i)T - 8.87e24iT^{2} \) |
| 89 | \( 1 + 6.17e12iT - 2.19e25T^{2} \) |
| 97 | \( 1 + 4.68e12T + 6.73e25T^{2} \) |
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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
−15.99109785708743308532065897910, −15.46136837888685929936094106903, −13.76908834033856347383554370814, −11.39106416303115251897199967758, −10.30143243866581864750569665432, −9.229786930626711775196181006346, −6.86274276996307423353203501685, −5.54523036861692323417358904745, −4.30090350667297323087316800601, −0.47861118784644370996406140413,
0.900311695348686638962458485171, 2.19001466329150210901052731783, 5.06378783553280544960006403550, 6.78684551647745793048191370456, 8.455838776304165910382003446999, 10.46290399693472273332166097801, 11.73156879676566519837322315698, 12.56696790638628444595137918547, 13.60395311066141452675026877358, 16.39358587891200299972900832027