| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (−74.2 + 42.8i)5-s + (282. − 488. i)7-s − 181. i·8-s + 484.·10-s + (1.25e3 + 726. i)11-s + (−420. − 728. i)13-s + (−2.76e3 + 1.59e3i)14-s + (−512. + 886. i)16-s + 5.09e3i·17-s − 6.68e3·19-s + (−2.37e3 − 1.37e3i)20-s + (−4.11e3 − 7.12e3i)22-s + (1.02e4 − 5.92e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.593 + 0.342i)5-s + (0.822 − 1.42i)7-s − 0.353i·8-s + 0.484·10-s + (0.945 + 0.546i)11-s + (−0.191 − 0.331i)13-s + (−1.00 + 0.581i)14-s + (−0.125 + 0.216i)16-s + 1.03i·17-s − 0.975·19-s + (−0.296 − 0.171i)20-s + (−0.386 − 0.668i)22-s + (0.843 − 0.487i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0871 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.302090462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302090462\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (74.2 - 42.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-282. + 488. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.25e3 - 726. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (420. + 728. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 5.09e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 6.68e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.02e4 + 5.92e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.33e4 - 7.71e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.65e4 + 2.86e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 8.43e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.39e4 + 8.07e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.30e4 + 5.72e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (1.37e5 + 7.95e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.50e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-1.66e5 + 9.60e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.91e5 + 3.31e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.18e5 + 2.05e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 6.88e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.79e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.85e5 + 6.67e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.54e5 + 3.77e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 8.71e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (6.06e5 - 1.05e6i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.12939474937241159208137322485, −10.75309535094951869885181151134, −9.606207867733437252456936051650, −8.284836153221400470451652722621, −7.50228612750807387569483588332, −6.56954565421154869881773463970, −4.48534067846225102935312211281, −3.66731207182021767791013759363, −1.81361539811072551388989294444, −0.56595895653891365981176091996,
1.08298386799578489664757642043, 2.55467492832781708549310436732, 4.45456975142607825735452786160, 5.60674609027488693476242914248, 6.79715088763131820240471412409, 8.150306349926323299296548484389, 8.757621461628610983619589577193, 9.607804023870496798549804627127, 11.36858567894546070420781884854, 11.58440149009380990007110024219
