| L(s) = 1 | + (209. − 121. i)2-s + (−1.56e3 + 1.52e3i)3-s + (2.11e4 − 3.66e4i)4-s + (−6.97e4 − 4.02e4i)5-s + (−1.43e5 + 5.09e5i)6-s + (−3.76e5 − 6.51e5i)7-s − 6.26e6i·8-s + (1.27e5 − 4.78e6i)9-s − 1.95e7·10-s + (1.49e7 − 8.64e6i)11-s + (2.27e7 + 8.95e7i)12-s + (−2.09e7 + 3.63e7i)13-s + (−1.57e8 − 9.11e7i)14-s + (1.70e8 − 4.33e7i)15-s + (−4.12e8 − 7.14e8i)16-s + 4.35e8i·17-s + ⋯ |
| L(s) = 1 | + (1.63 − 0.946i)2-s + (−0.716 + 0.697i)3-s + (1.28 − 2.23i)4-s + (−0.892 − 0.515i)5-s + (−0.513 + 1.82i)6-s + (−0.457 − 0.791i)7-s − 2.98i·8-s + (0.0266 − 0.999i)9-s − 1.95·10-s + (0.768 − 0.443i)11-s + (0.634 + 2.50i)12-s + (−0.334 + 0.579i)13-s + (−1.49 − 0.864i)14-s + (0.999 − 0.253i)15-s + (−1.53 − 2.66i)16-s + 1.06i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.873203 - 2.50231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.873203 - 2.50231i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.56e3 - 1.52e3i)T \) |
| good | 2 | \( 1 + (-209. + 121. i)T + (8.19e3 - 1.41e4i)T^{2} \) |
| 5 | \( 1 + (6.97e4 + 4.02e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 7 | \( 1 + (3.76e5 + 6.51e5i)T + (-3.39e11 + 5.87e11i)T^{2} \) |
| 11 | \( 1 + (-1.49e7 + 8.64e6i)T + (1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + (2.09e7 - 3.63e7i)T + (-1.96e15 - 3.40e15i)T^{2} \) |
| 17 | \( 1 - 4.35e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 9.97e8T + 7.99e17T^{2} \) |
| 23 | \( 1 + (1.81e9 + 1.04e9i)T + (5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + (-2.31e10 + 1.33e10i)T + (1.48e20 - 2.57e20i)T^{2} \) |
| 31 | \( 1 + (-1.04e10 + 1.80e10i)T + (-3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 - 5.64e10T + 9.01e21T^{2} \) |
| 41 | \( 1 + (1.74e11 + 1.00e11i)T + (1.89e22 + 3.28e22i)T^{2} \) |
| 43 | \( 1 + (-2.99e10 - 5.19e10i)T + (-3.69e22 + 6.39e22i)T^{2} \) |
| 47 | \( 1 + (1.62e11 - 9.38e10i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 - 5.89e11iT - 1.37e24T^{2} \) |
| 59 | \( 1 + (2.04e12 + 1.17e12i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (2.24e11 + 3.88e11i)T + (-4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (3.74e12 - 6.49e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 + 4.02e12iT - 8.27e25T^{2} \) |
| 73 | \( 1 - 1.50e13T + 1.22e26T^{2} \) |
| 79 | \( 1 + (-1.58e13 - 2.73e13i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 + (-8.14e12 + 4.69e12i)T + (3.68e26 - 6.37e26i)T^{2} \) |
| 89 | \( 1 + 5.29e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + (-3.97e12 - 6.87e12i)T + (-3.26e27 + 5.65e27i)T^{2} \) |
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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
−16.61627112618728444070178720861, −15.47248982224147807739332511887, −13.97182339327720189448630136980, −12.29756204727254545293300268813, −11.47165388180736007337023642402, −10.03991070633953807612087352097, −6.29561009740764757838775463790, −4.49352194135837073470159882466, −3.67083535370870773269116806951, −0.801015958194803226780517292383,
3.04428167831317185017049113938, 5.04399707339735943621810089616, 6.54458249968133271539747557950, 7.62233748720440814083983298546, 11.71640676910544974889806425005, 12.30450993146935838841743571176, 13.89990039342781209863661977187, 15.34960828132517427065536412447, 16.30648101386118054196510328319, 17.97115649552321623645964740648
