| L(s) = 1 | + (−7.13 − 8.78i)2-s + (15.6 − 37.8i)3-s + (−26.2 + 125. i)4-s + (221. − 91.7i)5-s + (−444. + 132. i)6-s + (1.08e3 − 1.08e3i)7-s + (1.28e3 − 662. i)8-s + (357. + 357. i)9-s + (−2.38e3 − 1.29e3i)10-s + (368. + 888. i)11-s + (4.33e3 + 2.96e3i)12-s + (−6.39e3 − 2.64e3i)13-s + (−1.72e4 − 1.79e3i)14-s − 9.82e3i·15-s + (−1.50e4 − 6.58e3i)16-s − 5.31e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.630 − 0.776i)2-s + (0.335 − 0.810i)3-s + (−0.205 + 0.978i)4-s + (0.792 − 0.328i)5-s + (−0.840 + 0.250i)6-s + (1.19 − 1.19i)7-s + (0.889 − 0.457i)8-s + (0.163 + 0.163i)9-s + (−0.754 − 0.408i)10-s + (0.0833 + 0.201i)11-s + (0.723 + 0.494i)12-s + (−0.807 − 0.334i)13-s + (−1.68 − 0.174i)14-s − 0.751i·15-s + (−0.915 − 0.401i)16-s − 0.262i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.703141 - 1.53814i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.703141 - 1.53814i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.13 + 8.78i)T \) |
| good | 3 | \( 1 + (-15.6 + 37.8i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-221. + 91.7i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-1.08e3 + 1.08e3i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-368. - 888. i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (6.39e3 + 2.64e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 5.31e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.56e3 + 649. i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (4.36e4 + 4.36e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (4.12e4 - 9.95e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-4.87e5 + 2.02e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (5.66e5 + 5.66e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-2.07e5 - 5.00e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.04e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (3.07e5 + 7.43e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (4.04e5 - 1.67e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (5.77e5 - 1.39e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.44e6 - 3.49e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-3.66e6 + 3.66e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-7.67e5 - 7.67e5i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 6.71e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-6.38e6 - 2.64e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (2.32e6 - 2.32e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 1.71e6T + 8.07e13T^{2} \) |
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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
−14.29364899244364914442110776328, −13.44140131667464912132832515713, −12.39764365564752684382654424830, −10.89203218469078325698636001318, −9.787768852593723647675299880098, −8.143115797947194231222727269149, −7.26060244416678565696081005717, −4.57683479205488457270771196944, −2.15519239555716872423606951481, −1.04103528605211931570097141386,
1.99527844188954022971441879092, 4.80359662327289784660287294459, 6.13624110568660399387504340775, 8.051872978695595756346022657147, 9.276687729446973209797577398325, 10.11957443069514870590345294878, 11.68733537996085956857136144320, 13.88922016978978516028293622847, 14.89388149505819991501198095405, 15.42669284352646154792145168731
