| L(s) = 1 | + (2 − 3.46i)2-s + (−2.72 + 4.72i)3-s + (−7.99 − 13.8i)4-s + (−41.7 + 72.3i)5-s + (10.9 + 18.8i)6-s − 105.·7-s − 63.9·8-s + (106. + 184. i)9-s + (166. + 289. i)10-s − 245.·11-s + 87.2·12-s + (95.4 + 165. i)13-s + (−211. + 366. i)14-s + (−227. − 394. i)15-s + (−128 + 221. i)16-s + (44.7 − 77.4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.174 + 0.302i)3-s + (−0.249 − 0.433i)4-s + (−0.746 + 1.29i)5-s + (0.123 + 0.214i)6-s − 0.816·7-s − 0.353·8-s + (0.438 + 0.760i)9-s + (0.528 + 0.914i)10-s − 0.611·11-s + 0.174·12-s + (0.156 + 0.271i)13-s + (−0.288 + 0.499i)14-s + (−0.261 − 0.452i)15-s + (−0.125 + 0.216i)16-s + (0.0375 − 0.0650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.514591 + 0.685092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514591 + 0.685092i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 19 | \( 1 + (-109. - 1.56e3i)T \) |
| good | 3 | \( 1 + (2.72 - 4.72i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (41.7 - 72.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + 105.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 245.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-95.4 - 165. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-44.7 + 77.4i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (1.91e3 + 3.31e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.05e3 + 1.82e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 - 4.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.03e4 - 1.79e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.10e3 + 1.91e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.00e4 - 1.74e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.90e4 - 3.30e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.22e3 + 3.86e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.34e4 - 2.33e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.90e3 - 5.02e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.52e3 - 4.37e3i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-1.82e4 + 3.16e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.51e4 + 7.81e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 9.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.34e4 + 4.05e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-8.08e3 + 1.40e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−15.55392161830492627857536596584, −14.42944090768874541255930374148, −13.20317562364886201786784125890, −11.87520501570984544181712609710, −10.68016355376099793068686353581, −10.00817851231574449733072063070, −7.81912026798950291954271029097, −6.27071276433167413059976697433, −4.22263824758429067801024711672, −2.75566889986901798942090679571,
0.44111127808638808496798053669, 3.81236536584864477489035446405, 5.39321988448181505915729968697, 6.99341878588908051240404940588, 8.340200419048235469688924656322, 9.615236703134677380097957738324, 11.78275971777423213504606478998, 12.72384648293756968365428463521, 13.43243030392477721651837419372, 15.43814787620242112029514372498
