| 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 | \( 1 + (2.04e5 + 1.48e5i)T + (2.57e11 + 7.92e11i)T^{2} \) |
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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
−15.53260175768315248011801227820, −13.91862110544964010050009438732, −12.76362772771623067458205532706, −11.37915690673663727596884094007, −9.421907755005420481797294935957, −8.373878804850454152705442925246, −7.09514453121504104670234734758, −5.95022087290524632770934497666, −3.06277717707968386854622020931, −0.18638673232412763346555552118,
2.11736253608773931398195833837, 4.04246331701111150447800688529, 6.29482051932689255888999948467, 8.450727405301905511376297373823, 9.741721618819831779548327484649, 10.46018103160170490400623867089, 11.79844378764694137360372429498, 13.10827216048985566135631122266, 14.77805510190377685004815830782, 15.72710456249550814958074191758
