| L(s) = 1 | + (64 − 110. i)2-s + (2.37e3 + 4.11e3i)3-s + (−8.19e3 − 1.41e4i)4-s + (1.10e5 − 1.91e5i)5-s + 6.08e5·6-s + (−2.15e6 − 3.33e5i)7-s − 2.09e6·8-s + (−4.14e6 + 7.17e6i)9-s + (−1.41e7 − 2.45e7i)10-s + (−3.11e7 − 5.39e7i)11-s + (3.89e7 − 6.75e7i)12-s + 3.80e8·13-s + (−1.74e8 + 2.17e8i)14-s + 1.05e9·15-s + (−1.34e8 + 2.32e8i)16-s + (−1.30e9 − 2.26e9i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.627 + 1.08i)3-s + (−0.249 − 0.433i)4-s + (0.633 − 1.09i)5-s + 0.888·6-s + (−0.988 − 0.152i)7-s − 0.353·8-s + (−0.288 + 0.499i)9-s + (−0.447 − 0.775i)10-s + (−0.481 − 0.834i)11-s + (0.313 − 0.543i)12-s + 1.68·13-s + (−0.442 + 0.551i)14-s + 1.58·15-s + (−0.125 + 0.216i)16-s + (−0.774 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.0899 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.69639 - 1.85651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69639 - 1.85651i\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-64 + 110. i)T \) |
| 7 | \( 1 + (2.15e6 + 3.33e5i)T \) |
| good | 3 | \( 1 + (-2.37e3 - 4.11e3i)T + (-7.17e6 + 1.24e7i)T^{2} \) |
| 5 | \( 1 + (-1.10e5 + 1.91e5i)T + (-1.52e10 - 2.64e10i)T^{2} \) |
| 11 | \( 1 + (3.11e7 + 5.39e7i)T + (-2.08e15 + 3.61e15i)T^{2} \) |
| 13 | \( 1 - 3.80e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + (1.30e9 + 2.26e9i)T + (-1.43e18 + 2.47e18i)T^{2} \) |
| 19 | \( 1 + (-9.84e8 + 1.70e9i)T + (-7.59e18 - 1.31e19i)T^{2} \) |
| 23 | \( 1 + (1.98e9 - 3.43e9i)T + (-1.33e20 - 2.30e20i)T^{2} \) |
| 29 | \( 1 - 3.43e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + (1.45e11 + 2.51e11i)T + (-1.17e22 + 2.03e22i)T^{2} \) |
| 37 | \( 1 + (-9.34e10 + 1.61e11i)T + (-1.66e23 - 2.88e23i)T^{2} \) |
| 41 | \( 1 - 1.45e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 8.85e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + (2.46e12 - 4.27e12i)T + (-6.03e24 - 1.04e25i)T^{2} \) |
| 53 | \( 1 + (-5.42e12 - 9.40e12i)T + (-3.65e25 + 6.33e25i)T^{2} \) |
| 59 | \( 1 + (-3.37e12 - 5.84e12i)T + (-1.82e26 + 3.16e26i)T^{2} \) |
| 61 | \( 1 + (9.55e12 - 1.65e13i)T + (-3.01e26 - 5.21e26i)T^{2} \) |
| 67 | \( 1 + (1.25e13 + 2.17e13i)T + (-1.23e27 + 2.13e27i)T^{2} \) |
| 71 | \( 1 + 2.23e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + (-3.55e13 - 6.16e13i)T + (-4.45e27 + 7.71e27i)T^{2} \) |
| 79 | \( 1 + (4.93e13 - 8.54e13i)T + (-1.45e28 - 2.52e28i)T^{2} \) |
| 83 | \( 1 + 2.30e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + (9.04e13 - 1.56e14i)T + (-8.70e28 - 1.50e29i)T^{2} \) |
| 97 | \( 1 - 9.75e14T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.71458608531092976832766144302, −13.73306017556056753198132298297, −13.09661196897637000603974782115, −11.04418890770724317210224094006, −9.540111200809356757105432731932, −8.890931502418824631060348121420, −5.79026735975207428383485167587, −4.22799621950994573936458108626, −2.92604381055614252122858677296, −0.76071335532978005161788703044,
1.93186502578903938223273350773, 3.35844907582419983529515800745, 6.17762818557477769294268338117, 6.94693081630093910407718427127, 8.522639689392289158815063873700, 10.42715779065508081900690495065, 12.74487421198414535403253878550, 13.47095805500065816011623682178, 14.63371860972982749374509136227, 15.96813523901517390566251733849
