| L(s) = 1 | + (8 + 13.8i)2-s + (86.7 − 150. i)3-s + (−127. + 221. i)4-s + (−1.17e3 − 2.03e3i)5-s + 2.77e3·6-s + (−6.34e3 + 246. i)7-s − 4.09e3·8-s + (−5.20e3 − 9.01e3i)9-s + (1.87e4 − 3.25e4i)10-s + (3.55e4 − 6.15e4i)11-s + (2.22e4 + 3.84e4i)12-s + 8.94e4·13-s + (−5.41e4 − 8.59e4i)14-s − 4.06e5·15-s + (−3.27e4 − 5.67e4i)16-s + (−1.61e4 + 2.79e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.618 − 1.07i)3-s + (−0.249 + 0.433i)4-s + (−0.839 − 1.45i)5-s + 0.874·6-s + (−0.999 + 0.0387i)7-s − 0.353·8-s + (−0.264 − 0.457i)9-s + (0.593 − 1.02i)10-s + (0.732 − 1.26i)11-s + (0.309 + 0.535i)12-s + 0.868·13-s + (−0.377 − 0.598i)14-s − 2.07·15-s + (−0.125 − 0.216i)16-s + (−0.0468 + 0.0812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0246 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0246 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.18555 - 1.21512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18555 - 1.21512i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-8 - 13.8i)T \) |
| 7 | \( 1 + (6.34e3 - 246. i)T \) |
| good | 3 | \( 1 + (-86.7 + 150. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.17e3 + 2.03e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (-3.55e4 + 6.15e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 8.94e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (1.61e4 - 2.79e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-3.89e5 - 6.74e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (8.86e5 + 1.53e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 2.12e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-2.15e6 + 3.73e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (3.10e6 + 5.37e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 6.40e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.99e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.75e7 - 3.03e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.13e7 + 3.69e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.45e7 + 2.51e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (3.09e7 + 5.35e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.00e8 - 1.73e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 - 3.03e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.45e8 + 2.52e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-5.44e7 - 9.42e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.16e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-9.32e7 - 1.61e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 9.53e8T + 7.60e17T^{2} \) |
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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
−16.58381326434673909243700963516, −16.02754227604660677413885046388, −13.99818466741991222204938684393, −12.99584086558268369700118339536, −12.05965097190921562683832018708, −8.851368649284368415064024725890, −7.967126662607652403091882262658, −6.14573766690199134306524848058, −3.74479079233610291051878332201, −0.811276141623970159815430949866,
3.06484796766458959250157072646, 4.00689270866333608640289995398, 6.89142093130583675919529851951, 9.384576392835783561870823139684, 10.44682153521670058716181759748, 11.84828559376463732881762898120, 13.84148501677535983820959804861, 15.16465242924347883440117381417, 15.70044176043329077880353555917, 18.11155513093879023743681790081
