| L(s) = 1 | + (−4 + 6.92i)2-s + (38.4 + 66.5i)3-s + (−31.9 − 55.4i)4-s + (10.6 − 18.3i)5-s − 615.·6-s + (−642. + 641. i)7-s + 511.·8-s + (−1.86e3 + 3.22e3i)9-s + (84.9 + 147. i)10-s + (−1.63e3 − 2.82e3i)11-s + (2.46e3 − 4.26e3i)12-s + 1.02e4·13-s + (−1.87e3 − 7.01e3i)14-s + 1.63e3·15-s + (−2.04e3 + 3.54e3i)16-s + (1.82e4 + 3.16e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.821 + 1.42i)3-s + (−0.249 − 0.433i)4-s + (0.0380 − 0.0658i)5-s − 1.16·6-s + (−0.707 + 0.706i)7-s + 0.353·8-s + (−0.851 + 1.47i)9-s + (0.0268 + 0.0465i)10-s + (−0.369 − 0.640i)11-s + (0.410 − 0.711i)12-s + 1.29·13-s + (−0.182 − 0.683i)14-s + 0.124·15-s + (−0.125 + 0.216i)16-s + (0.901 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.508429 + 1.34475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.508429 + 1.34475i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 7 | \( 1 + (642. - 641. i)T \) |
| good | 3 | \( 1 + (-38.4 - 66.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-10.6 + 18.3i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (1.63e3 + 2.82e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.82e4 - 3.16e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.62e4 + 2.80e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (8.37e3 - 1.44e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 4.39e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (6.69e4 + 1.15e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.89e5 + 5.02e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 5.32e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.03e5 + 1.79e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-5.44e5 - 9.42e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (8.08e4 + 1.39e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.22e5 + 7.31e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (6.13e5 + 1.06e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (8.19e5 + 1.41e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.49e6 + 4.32e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 3.38e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.26e6 - 5.66e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.35e6T + 8.07e13T^{2} \) |
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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
−18.59146625388930304287814459632, −16.63767789141504185704496875293, −15.75067220321697618511520094467, −14.90841791928877564351379723208, −13.37593109711960404682013509539, −10.74964288509576927491975356349, −9.378827565290466004694295664174, −8.401395713794694159538445818033, −5.69933867205588996743180483269, −3.49235206994215538916097971359,
1.10464563684626396521844490985, 3.08486100807694147698516493363, 6.90797301692274462850428231176, 8.220474736804353486517114638871, 9.964668166660463024184031814782, 12.02578565215134253898458953301, 13.22334667468482983806600491608, 14.09016188798609979329841345143, 16.35152115238444363179806765882, 18.22178061001384319633236580877
