| L(s) = 1 | + (−5.06 − 6.96i)2-s + (4.81 + 14.8i)3-s + (−3.15 + 9.71i)4-s + (−8.79 − 6.38i)5-s + (78.9 − 108. i)6-s + (−242. − 78.6i)7-s + (−440. + 143. i)8-s + (−196. + 142. i)9-s + 93.6i·10-s + (−153. + 1.32e3i)11-s − 159.·12-s + (617. + 850. i)13-s + (677. + 2.08e3i)14-s + (52.3 − 161. i)15-s + (3.75e3 + 2.73e3i)16-s + (−5.01e3 + 6.90e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.871i)2-s + (0.178 + 0.549i)3-s + (−0.0493 + 0.151i)4-s + (−0.0703 − 0.0511i)5-s + (0.365 − 0.502i)6-s + (−0.705 − 0.229i)7-s + (−0.860 + 0.279i)8-s + (−0.269 + 0.195i)9-s + 0.0936i·10-s + (−0.115 + 0.993i)11-s − 0.0921·12-s + (0.281 + 0.387i)13-s + (0.246 + 0.760i)14-s + (0.0155 − 0.0477i)15-s + (0.917 + 0.666i)16-s + (−1.02 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.191845 + 0.251632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191845 + 0.251632i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.81 - 14.8i)T \) |
| 11 | \( 1 + (153. - 1.32e3i)T \) |
| good | 2 | \( 1 + (5.06 + 6.96i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (8.79 + 6.38i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (242. + 78.6i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (-617. - 850. i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (5.01e3 - 6.90e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (1.82e3 - 594. i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 + 1.04e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.51e4 + 4.92e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (-2.36e4 + 1.71e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (3.32e3 - 1.02e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-3.33e3 + 1.08e3i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 - 4.83e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (4.38e4 + 1.34e5i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-7.49e4 + 5.44e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (7.37e4 - 2.27e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.15e4 - 1.58e4i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 2.98e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-2.95e5 - 2.14e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (6.79e5 + 2.20e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (2.50e5 + 3.44e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (4.12e5 - 5.67e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 - 4.28e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + (2.04e5 - 1.48e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.72710456249550814958074191758, −14.77805510190377685004815830782, −13.10827216048985566135631122266, −11.79844378764694137360372429498, −10.46018103160170490400623867089, −9.741721618819831779548327484649, −8.450727405301905511376297373823, −6.29482051932689255888999948467, −4.04246331701111150447800688529, −2.11736253608773931398195833837,
0.18638673232412763346555552118, 3.06277717707968386854622020931, 5.95022087290524632770934497666, 7.09514453121504104670234734758, 8.373878804850454152705442925246, 9.421907755005420481797294935957, 11.37915690673663727596884094007, 12.76362772771623067458205532706, 13.91862110544964010050009438732, 15.53260175768315248011801227820
