| L(s) = 1 | + (−11.5 + 24.4i)3-s + (−113. + 65.3i)5-s + (−143. + 248. i)7-s + (−463. − 562. i)9-s + (202. + 116. i)11-s + (217. + 375. i)13-s + (−293. − 3.51e3i)15-s − 2.28e3i·17-s − 740.·19-s + (−4.42e3 − 6.37e3i)21-s + (1.86e4 − 1.07e4i)23-s + (740. − 1.28e3i)25-s + (1.90e4 − 4.85e3i)27-s + (−2.31e4 − 1.33e4i)29-s + (−2.46e4 − 4.26e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.426 + 0.904i)3-s + (−0.906 + 0.523i)5-s + (−0.418 + 0.725i)7-s + (−0.636 − 0.771i)9-s + (0.151 + 0.0877i)11-s + (0.0987 + 0.171i)13-s + (−0.0868 − 1.04i)15-s − 0.465i·17-s − 0.108·19-s + (−0.477 − 0.687i)21-s + (1.53 − 0.883i)23-s + (0.0473 − 0.0820i)25-s + (0.969 − 0.246i)27-s + (−0.947 − 0.547i)29-s + (−0.827 − 1.43i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00837 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.00837 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.101716 - 0.102572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.101716 - 0.102572i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (11.5 - 24.4i)T \) |
| good | 5 | \( 1 + (113. - 65.3i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (143. - 248. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-202. - 116. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-217. - 375. i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 2.28e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 740.T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.86e4 + 1.07e4i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.31e4 + 1.33e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (2.46e4 + 4.26e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 8.14e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + (2.51e4 - 1.45e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (8.42e3 - 1.45e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (9.91e4 + 5.72e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 6.21e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-3.15e5 + 1.82e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.17e5 - 3.76e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.02e5 - 1.78e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 1.12e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.58e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (3.68e5 - 6.38e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.78e5 + 1.03e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 1.08e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-5.73e5 + 9.92e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97092732780157889849948812906, −11.68590050863206982741676586978, −11.11402218599430167878330864003, −9.752594653764676606818501085157, −8.731131900320478663805451936648, −7.10531873450463993143917121579, −5.73190084876667395866699073407, −4.25181632448035424247178680045, −2.97140681405999097820635690163, −0.06515236238281562465272213987,
1.26227394388011345481907961350, 3.52865666690143811568595810540, 5.17773469049710083202168370829, 6.76932452582724937756420128019, 7.68378142930493114618452710045, 8.875533508941013613012739079583, 10.64116747432676664127563759526, 11.59381729235994470995987580021, 12.68421236659415026129060596072, 13.33441463315559567860201820466
