| L(s) = 1 | + (195. + 112. i)2-s + (840. + 2.01e3i)3-s + (1.72e4 + 2.98e4i)4-s + (1.03e5 − 5.96e4i)5-s + (−6.34e4 + 4.89e5i)6-s + (1.63e5 − 2.82e5i)7-s + 4.07e6i·8-s + (−3.36e6 + 3.39e6i)9-s + 2.69e7·10-s + (−2.55e7 − 1.47e7i)11-s + (−4.57e7 + 5.99e7i)12-s + (−3.84e7 − 6.66e7i)13-s + (6.37e7 − 3.68e7i)14-s + (2.07e8 + 1.58e8i)15-s + (−1.77e8 + 3.07e8i)16-s + 3.47e8i·17-s + ⋯ |
| L(s) = 1 | + (1.52 + 0.880i)2-s + (0.384 + 0.923i)3-s + (1.05 + 1.82i)4-s + (1.32 − 0.763i)5-s + (−0.226 + 1.74i)6-s + (0.198 − 0.343i)7-s + 1.94i·8-s + (−0.704 + 0.709i)9-s + 2.69·10-s + (−1.30 − 0.756i)11-s + (−1.27 + 1.67i)12-s + (−0.612 − 1.06i)13-s + (0.604 − 0.349i)14-s + (1.21 + 0.927i)15-s + (−0.661 + 1.14i)16-s + 0.847i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0909 - 0.995i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.0909 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(3.34736 + 3.66717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.34736 + 3.66717i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-840. - 2.01e3i)T \) |
| good | 2 | \( 1 + (-195. - 112. i)T + (8.19e3 + 1.41e4i)T^{2} \) |
| 5 | \( 1 + (-1.03e5 + 5.96e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 7 | \( 1 + (-1.63e5 + 2.82e5i)T + (-3.39e11 - 5.87e11i)T^{2} \) |
| 11 | \( 1 + (2.55e7 + 1.47e7i)T + (1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + (3.84e7 + 6.66e7i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 17 | \( 1 - 3.47e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 1.79e7T + 7.99e17T^{2} \) |
| 23 | \( 1 + (-1.21e9 + 7.03e8i)T + (5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-8.71e9 - 5.02e9i)T + (1.48e20 + 2.57e20i)T^{2} \) |
| 31 | \( 1 + (-6.77e9 - 1.17e10i)T + (-3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 - 4.16e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (2.44e11 - 1.40e11i)T + (1.89e22 - 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-1.29e9 + 2.23e9i)T + (-3.69e22 - 6.39e22i)T^{2} \) |
| 47 | \( 1 + (3.91e11 + 2.25e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 - 1.74e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (-3.57e12 + 2.06e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (8.17e11 - 1.41e12i)T + (-4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (-1.15e12 - 1.99e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 8.75e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 + 1.12e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + (5.16e11 - 8.94e11i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-8.74e12 - 5.04e12i)T + (3.68e26 + 6.37e26i)T^{2} \) |
| 89 | \( 1 + 5.62e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (5.37e13 - 9.30e13i)T + (-3.26e27 - 5.65e27i)T^{2} \) |
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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
−17.31939043323578982026002903140, −16.29418286899494925973067826901, −15.03617345412608706833029953646, −13.75506854012508221512846723276, −12.92063905642656203533140346814, −10.34619815883595959681250023771, −8.211607669544185305354623994205, −5.70767383156000494811910763114, −4.86075769374670566577568872545, −2.89061606907897603532520799228,
1.99283882169756449365022618688, 2.65293672947522432583563551595, 5.27473065633946667079011339768, 6.80015540137940149226736970449, 9.940149632788496618274411367811, 11.68747410303414935082711980009, 13.08344954822003109201505300422, 13.94283825185851819448987135776, 14.93846463895461723759972580136, 17.88508947972877684886506441660
