| L(s) = 1 | + (−16 + 27.7i)2-s + (−511. − 886. i)4-s + (6.10e3 + 1.05e4i)5-s + (6.94e3 − 1.20e4i)7-s + 3.27e4·8-s − 3.90e5·10-s + (−1.66e5 + 2.88e5i)11-s + (9.18e5 + 1.59e6i)13-s + (2.22e5 + 3.84e5i)14-s + (−5.24e5 + 9.08e5i)16-s + 6.76e6·17-s + 1.31e7·19-s + (6.25e6 − 1.08e7i)20-s + (−5.33e6 − 9.23e6i)22-s + (−1.47e7 − 2.54e7i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.873 + 1.51i)5-s + (0.156 − 0.270i)7-s + 0.353·8-s − 1.23·10-s + (−0.311 + 0.540i)11-s + (0.686 + 1.18i)13-s + (0.110 + 0.191i)14-s + (−0.125 + 0.216i)16-s + 1.15·17-s + 1.21·19-s + (0.436 − 0.756i)20-s + (−0.220 − 0.382i)22-s + (−0.476 − 0.825i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.577015941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.577015941\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (16 - 27.7i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-6.10e3 - 1.05e4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-6.94e3 + 1.20e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (1.66e5 - 2.88e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (-9.18e5 - 1.59e6i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 - 6.76e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.31e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + (1.47e7 + 2.54e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (1.25e7 - 2.16e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-1.23e8 - 2.13e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 - 4.11e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (2.97e8 + 5.14e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (4.76e8 - 8.25e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (-3.02e8 + 5.24e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 - 5.75e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (9.96e8 + 1.72e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-5.04e9 + 8.74e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (4.61e9 + 7.99e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 - 2.20e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.58e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-5.02e9 + 8.69e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (-2.18e10 + 3.78e10i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 - 9.65e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (2.36e10 - 4.09e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−10.85216531458849331816518816253, −10.12042523419420516042670627514, −9.361023294407195719323409253059, −7.920468148768569697252979278784, −6.94770023166833007011577128448, −6.30958173655257373097495503382, −5.15301366795962659828814139056, −3.59051185064717311499716888239, −2.32563026421510874534017372218, −1.17801894876843414281125874864,
0.73152075774113750338068220137, 1.15586812387611084936561482600, 2.50862426990483494070138726438, 3.79888689562668241802610035601, 5.35709479777994763154902387954, 5.70077867745050757677550342284, 7.86145306009737231096136135230, 8.462383609976205136419459262219, 9.565846348776641417540985802698, 10.13007793806867440848023804034
