| L(s) = 1 | + (3.56 + 4.46i)2-s + (18.2 − 22.8i)3-s + (−0.152 + 0.667i)4-s + (−27.3 + 13.1i)5-s + 166.·6-s − 9.91·7-s + (161. − 77.6i)8-s + (−135. − 594. i)9-s + (−156. − 75.4i)10-s + (51.8 + 227. i)11-s + (12.4 + 15.6i)12-s + (41.6 − 20.0i)13-s + (−35.3 − 44.3i)14-s + (−197. + 865. i)15-s + (941. + 453. i)16-s + (869. + 418. i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.790i)2-s + (1.16 − 1.46i)3-s + (−0.00476 + 0.0208i)4-s + (−0.489 + 0.235i)5-s + 1.89·6-s − 0.0764·7-s + (0.891 − 0.429i)8-s + (−0.557 − 2.44i)9-s + (−0.495 − 0.238i)10-s + (0.129 + 0.566i)11-s + (0.0249 + 0.0313i)12-s + (0.0683 − 0.0329i)13-s + (−0.0481 − 0.0604i)14-s + (−0.226 + 0.992i)15-s + (0.919 + 0.442i)16-s + (0.729 + 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.89556 - 0.828402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.89556 - 0.828402i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.11e4 + 4.78e3i)T \) |
| good | 2 | \( 1 + (-3.56 - 4.46i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-18.2 + 22.8i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (27.3 - 13.1i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 9.91T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-51.8 - 227. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-41.6 + 20.0i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-869. - 418. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (164. - 722. i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-1.07e3 - 4.70e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (4.81e3 + 6.03e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-1.27e3 - 1.60e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 - 5.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-7.08e3 - 8.88e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-2.87e3 + 1.25e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-2.18e4 - 1.05e4i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (2.02e4 + 9.77e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.86e4 - 2.34e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (2.22e3 - 9.75e3i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-1.75e4 + 7.67e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (2.53e4 - 1.22e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 3.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (5.17e4 - 6.49e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.99e4 + 8.77e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (1.47e4 + 6.45e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.85319173898495324630475956698, −13.72108189260544249247058475295, −13.02697183672746012296044337385, −11.75045859880930336067903120212, −9.603812409168719822037984108457, −7.85500906464217846928181010508, −7.29984903660365487736347199169, −5.98256582314714457465930620388, −3.61726563945416771167107598168, −1.56385783842239635572219766719,
2.72303878485457215326233765514, 3.80794707373951856756804767420, 4.85873212813594697165934248485, 7.918660478191489386852081495284, 8.966672984673943637711023113533, 10.36474304957149055979259574462, 11.31129096772513452592509488498, 12.79337882418206506762285696102, 14.01455347521818076833023059675, 14.78511648852249898003338594280
