| L(s) = 1 | + (129. − 74.7i)2-s + (1.39e3 + 1.68e3i)3-s + (2.99e3 − 5.19e3i)4-s + (3.19e4 + 1.84e4i)5-s + (3.06e5 + 1.13e5i)6-s + (1.53e5 + 2.66e5i)7-s + 1.55e6i·8-s + (−8.71e5 + 4.70e6i)9-s + 5.51e6·10-s + (8.07e6 − 4.66e6i)11-s + (1.29e7 − 2.21e6i)12-s + (2.46e7 − 4.26e7i)13-s + (3.98e7 + 2.30e7i)14-s + (1.36e7 + 7.95e7i)15-s + (1.65e8 + 2.86e8i)16-s − 1.60e8i·17-s + ⋯ |
| L(s) = 1 | + (1.01 − 0.584i)2-s + (0.639 + 0.768i)3-s + (0.182 − 0.316i)4-s + (0.408 + 0.236i)5-s + (1.09 + 0.404i)6-s + (0.186 + 0.323i)7-s + 0.741i·8-s + (−0.182 + 0.983i)9-s + 0.551·10-s + (0.414 − 0.239i)11-s + (0.360 − 0.0619i)12-s + (0.392 − 0.680i)13-s + (0.377 + 0.218i)14-s + (0.0799 + 0.465i)15-s + (0.615 + 1.06i)16-s − 0.391i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(3.60653 + 0.949581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.60653 + 0.949581i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.39e3 - 1.68e3i)T \) |
| good | 2 | \( 1 + (-129. + 74.7i)T + (8.19e3 - 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-3.19e4 - 1.84e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-1.53e5 - 2.66e5i)T + (-3.39e11 + 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-8.07e6 + 4.66e6i)T + (1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + (-2.46e7 + 4.26e7i)T + (-1.96e15 - 3.40e15i)T^{2} \) |
| 17 | \( 1 + 1.60e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.10e9T + 7.99e17T^{2} \) |
| 23 | \( 1 + (5.63e9 + 3.25e9i)T + (5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-5.84e9 + 3.37e9i)T + (1.48e20 - 2.57e20i)T^{2} \) |
| 31 | \( 1 + (5.34e9 - 9.25e9i)T + (-3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + 1.45e11T + 9.01e21T^{2} \) |
| 41 | \( 1 + (-1.88e11 - 1.08e11i)T + (1.89e22 + 3.28e22i)T^{2} \) |
| 43 | \( 1 + (7.02e10 + 1.21e11i)T + (-3.69e22 + 6.39e22i)T^{2} \) |
| 47 | \( 1 + (-4.86e11 + 2.80e11i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + 2.07e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-1.90e12 - 1.10e12i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-2.01e12 - 3.48e12i)T + (-4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-1.82e12 + 3.15e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 - 4.60e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 1.45e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-1.41e13 - 2.44e13i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-3.75e12 + 2.16e12i)T + (3.68e26 - 6.37e26i)T^{2} \) |
| 89 | \( 1 + 1.98e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-8.91e12 - 1.54e13i)T + (-3.26e27 + 5.65e27i)T^{2} \) |
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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
−17.94258707026678454240548078277, −16.01610061530363804628626971526, −14.45258245170900582725138182825, −13.64651097630733584010497486187, −11.86892590693310508254565634915, −10.25059891829819961761343397932, −8.418190772541955834037814557974, −5.44189278214658252532809633932, −3.78756488303992858912325292319, −2.39772815165412572258372841887,
1.43329428885575592284035153870, 3.83396396273521991516762379028, 5.88864757449422782764480462712, 7.38773133368908369753453266148, 9.435232159417564532898080842019, 12.14699014339349488637130993715, 13.62271101576885812982070070064, 14.19931750041462232085433843741, 15.76344762182386141114706316384, 17.60253163597121221007878543171
