| L(s) = 1 | + (4.99 − 6.87i)2-s + (−4.81 + 14.8i)3-s + (−2.56 − 7.90i)4-s + (−1.33 + 0.967i)5-s + (77.9 + 107. i)6-s + (525. − 170. i)7-s + (450. + 146. i)8-s + (−196. − 142. i)9-s + 13.9i·10-s + (1.33e3 + 21.3i)11-s + 129.·12-s + (−962. + 1.32e3i)13-s + (1.45e3 − 4.47e3i)14-s + (−7.92 − 24.3i)15-s + (3.68e3 − 2.67e3i)16-s + (−336. − 463. i)17-s + ⋯ |
| L(s) = 1 | + (0.624 − 0.859i)2-s + (−0.178 + 0.549i)3-s + (−0.0401 − 0.123i)4-s + (−0.0106 + 0.00773i)5-s + (0.360 + 0.496i)6-s + (1.53 − 0.497i)7-s + (0.879 + 0.285i)8-s + (−0.269 − 0.195i)9-s + 0.0139i·10-s + (0.999 + 0.0160i)11-s + 0.0750·12-s + (−0.437 + 0.602i)13-s + (0.529 − 1.62i)14-s + (−0.00234 − 0.00722i)15-s + (0.900 − 0.654i)16-s + (−0.0685 − 0.0943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.52300 - 0.511123i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52300 - 0.511123i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.81 - 14.8i)T \) |
| 11 | \( 1 + (-1.33e3 - 21.3i)T \) |
| good | 2 | \( 1 + (-4.99 + 6.87i)T + (-19.7 - 60.8i)T^{2} \) |
| 5 | \( 1 + (1.33 - 0.967i)T + (4.82e3 - 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-525. + 170. i)T + (9.51e4 - 6.91e4i)T^{2} \) |
| 13 | \( 1 + (962. - 1.32e3i)T + (-1.49e6 - 4.59e6i)T^{2} \) |
| 17 | \( 1 + (336. + 463. i)T + (-7.45e6 + 2.29e7i)T^{2} \) |
| 19 | \( 1 + (8.38e3 + 2.72e3i)T + (3.80e7 + 2.76e7i)T^{2} \) |
| 23 | \( 1 - 1.84e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (3.00e4 - 9.76e3i)T + (4.81e8 - 3.49e8i)T^{2} \) |
| 31 | \( 1 + (2.39e4 + 1.73e4i)T + (2.74e8 + 8.44e8i)T^{2} \) |
| 37 | \( 1 + (1.97e4 + 6.07e4i)T + (-2.07e9 + 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-1.33e4 - 4.34e3i)T + (3.84e9 + 2.79e9i)T^{2} \) |
| 43 | \( 1 + 3.85e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (3.60e4 - 1.10e5i)T + (-8.72e9 - 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-2.75e4 - 2.00e4i)T + (6.84e9 + 2.10e10i)T^{2} \) |
| 59 | \( 1 + (9.65e4 + 2.97e5i)T + (-3.41e10 + 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.98e5 + 2.72e5i)T + (-1.59e10 + 4.89e10i)T^{2} \) |
| 67 | \( 1 + 7.49e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-6.36e3 + 4.62e3i)T + (3.95e10 - 1.21e11i)T^{2} \) |
| 73 | \( 1 + (3.35e5 - 1.08e5i)T + (1.22e11 - 8.89e10i)T^{2} \) |
| 79 | \( 1 + (4.47e5 - 6.15e5i)T + (-7.51e10 - 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-1.73e5 - 2.38e5i)T + (-1.01e11 + 3.10e11i)T^{2} \) |
| 89 | \( 1 - 3.34e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.38e4 - 1.00e4i)T + (2.57e11 + 7.92e11i)T^{2} \) |
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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
−14.91493504156700515487853046312, −14.17801467254916548995124765943, −12.74559312970941506765953402516, −11.26523740892783942271175678855, −11.04368638091732940546001830749, −9.093971357929892274999723785493, −7.36408047967569485980021607542, −4.94064528285296306053893317833, −3.91948543142429994596089540139, −1.75432002276118690233258372872,
1.61216921950091983478602613317, 4.63666572157312742772871375536, 5.88548664221286402541291888984, 7.27976287763988179631801775798, 8.549856734724410030924948549744, 10.72970942027364267103643216772, 11.97348797323544338697357327298, 13.32580986324023148103306133613, 14.75705119054180224035065715756, 14.89641690396445200426643538943
