| L(s) = 1 | + (4.5 + 7.79i)3-s + (−23.0 + 39.9i)5-s + (112. − 64.8i)7-s + (−40.5 + 70.1i)9-s + (315. + 546. i)11-s − 1.07e3·13-s − 415.·15-s + (−80.5 − 139. i)17-s + (−588. + 1.01e3i)19-s + (1.01e3 + 583. i)21-s + (−1.08e3 + 1.87e3i)23-s + (499. + 864. i)25-s − 729·27-s − 4.49e3·29-s + (−159. − 275. i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.412 + 0.714i)5-s + (0.866 − 0.500i)7-s + (−0.166 + 0.288i)9-s + (0.786 + 1.36i)11-s − 1.77·13-s − 0.476·15-s + (−0.0676 − 0.117i)17-s + (−0.373 + 0.647i)19-s + (0.500 + 0.288i)21-s + (−0.426 + 0.738i)23-s + (0.159 + 0.276i)25-s − 0.192·27-s − 0.991·29-s + (−0.0297 − 0.0515i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.709868 + 1.32480i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709868 + 1.32480i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (-112. + 64.8i)T \) |
| good | 5 | \( 1 + (23.0 - 39.9i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-315. - 546. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 1.07e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + (80.5 + 139. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (588. - 1.01e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.08e3 - 1.87e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 4.49e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (159. + 275. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (7.59e3 - 1.31e4i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 455.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.03e4 + 1.79e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.65e3 - 1.67e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (3.18e3 + 5.51e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.45e4 + 4.25e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.70e4 + 2.94e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 6.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.43e3 - 7.67e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.72e4 - 2.98e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.04e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-1.01e4 + 1.75e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 5.40e4T + 8.58e9T^{2} \) |
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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
−14.07675148872478654453447857927, −12.40306779388706635323732208957, −11.46719893314675671292244673145, −10.29674782473129323001954977895, −9.438423143405865236972338388396, −7.76275016492297043572347826177, −7.05698464223477029641445075135, −4.97142498976516659030371445022, −3.87601836887854677612210552990, −2.05863421924980949923692904651,
0.59708451868360033029805090933, 2.36035497088433317751405215993, 4.31797698229704536341713315901, 5.69917077417117844043269322620, 7.32389543320265269165928588100, 8.447653488393144275511144238186, 9.182087485484873618611260321011, 11.00711257016931531405157934910, 12.01495564874337591776574733795, 12.72890334177401793563532205446
