L(s) = 1 | + (90.9 − 90.9i)3-s + (−434. − 449. i)5-s + (−508. − 508. i)7-s − 9.98e3i·9-s + 7.02e3·11-s + (−8.07e3 + 8.07e3i)13-s + (−8.03e4 − 1.36e3i)15-s + (−1.02e5 − 1.02e5i)17-s − 5.95e4i·19-s − 9.24e4·21-s + (−1.32e5 + 1.32e5i)23-s + (−1.32e4 + 3.90e5i)25-s + (−3.11e5 − 3.11e5i)27-s + 3.92e5i·29-s + 5.07e5·31-s + ⋯ |
L(s) = 1 | + (1.12 − 1.12i)3-s + (−0.694 − 0.719i)5-s + (−0.211 − 0.211i)7-s − 1.52i·9-s + 0.479·11-s + (−0.282 + 0.282i)13-s + (−1.58 − 0.0269i)15-s + (−1.23 − 1.23i)17-s − 0.457i·19-s − 0.475·21-s + (−0.474 + 0.474i)23-s + (−0.0339 + 0.999i)25-s + (−0.586 − 0.586i)27-s + 0.554i·29-s + 0.549·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.181027 + 1.44766i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.181027 + 1.44766i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (434. + 449. i)T \) |
good | 3 | \( 1 + (-90.9 + 90.9i)T - 6.56e3iT^{2} \) |
| 7 | \( 1 + (508. + 508. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 - 7.02e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + (8.07e3 - 8.07e3i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (1.02e5 + 1.02e5i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + 5.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (1.32e5 - 1.32e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 - 3.92e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 5.07e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (6.10e4 + 6.10e4i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 1.81e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (1.47e6 - 1.47e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (5.66e6 - 5.66e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + 1.74e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.96e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-1.12e7 - 1.12e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 3.01e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.52e7 + 2.52e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 8.14e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.99e7 + 1.99e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 + 8.20e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-3.37e7 - 3.37e7i)T + 7.83e15iT^{2} \) |
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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
−12.34236556361344313449856633634, −11.42206632040376959549955648596, −9.409622658392384234958654466913, −8.652245134459299541960158929867, −7.55674193149675283150825002116, −6.71586389469267682971193455666, −4.61358367406642615657577747576, −3.14659660491641450849390886043, −1.72010963139522757146327657095, −0.36522932956607404120631907800,
2.36154472389570283169808709543, 3.56634750379456692411125756464, 4.40127653075212226832920838329, 6.40735563018568854949403902524, 7.962638503015045290129953618222, 8.813530310012713106290313253374, 10.01818010258391302866894969250, 10.81969327720687606076751367903, 12.16773951357810367538948246359, 13.64015559440327403292320052705