L(s) = 1 | + (−3 + 26.8i)3-s − 160. i·5-s − 242·7-s + (−711. − 160. i)9-s − 1.77e3i·11-s − 2.61e3·13-s + (4.32e3 + 482. i)15-s + 7.08e3i·17-s + 5.78e3·19-s + (726 − 6.49e3i)21-s + 9.33e3i·23-s − 1.02e4·25-s + (6.45e3 − 1.85e4i)27-s + 1.23e4i·29-s + 2.04e4·31-s + ⋯ |
L(s) = 1 | + (−0.111 + 0.993i)3-s − 1.28i·5-s − 0.705·7-s + (−0.975 − 0.220i)9-s − 1.33i·11-s − 1.19·13-s + (1.28 + 0.143i)15-s + 1.44i·17-s + 0.843·19-s + (0.0783 − 0.701i)21-s + 0.767i·23-s − 0.658·25-s + (0.327 − 0.944i)27-s + 0.508i·29-s + 0.686·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.054574243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.054574243\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (3 - 26.8i)T \) |
good | 5 | \( 1 + 160. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 242T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.77e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.61e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.08e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.78e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 9.33e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 1.23e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.04e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 4.67e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 3.54e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.86e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.12e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.71e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.47e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 8.43e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.24e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.13e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.59e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 5.15e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.25e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.99e5T + 8.32e11T^{2} \) |
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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
−11.69459031588614280336872636305, −10.56270965617148660508370892538, −9.570677948307797842425285873053, −8.908981435279535658268877350131, −7.918749533445129065592490459863, −6.08527598254909184821126892003, −5.27733038677848672352707777540, −4.15888718065234049936223401991, −3.01083742324964902341832456202, −0.920212159963125455458440220730,
0.36919960835207188313787989637, 2.29160517542408807612817766662, 2.96598959971481337731950518805, 4.88962426143364986960762175514, 6.37421571650861583044145999677, 7.11443070050069136174131212948, 7.65291288615962293471840684119, 9.453356106266141848253337074949, 10.14886051677067241700558332520, 11.44591770311912567291976852785