L(s) = 1 | + (−4.20e4 + 4.20e4i)2-s + (−2.23e7 − 2.23e7i)3-s + 7.62e8i·4-s + (−7.62e10 − 1.32e11i)5-s + 1.87e12·6-s + (−2.39e13 + 2.39e13i)7-s + (−2.12e14 − 2.12e14i)8-s − 8.55e14i·9-s + (8.75e15 + 2.35e15i)10-s − 3.17e16·11-s + (1.70e16 − 1.70e16i)12-s + (−5.32e17 − 5.32e17i)13-s − 2.01e18i·14-s + (−1.25e18 + 4.65e18i)15-s + 1.45e19·16-s + (4.10e19 − 4.10e19i)17-s + ⋯ |
L(s) = 1 | + (−0.641 + 0.641i)2-s + (−0.518 − 0.518i)3-s + 0.177i·4-s + (−0.499 − 0.866i)5-s + 0.665·6-s + (−0.721 + 0.721i)7-s + (−0.755 − 0.755i)8-s − 0.461i·9-s + (0.875 + 0.235i)10-s − 0.690·11-s + (0.0920 − 0.0920i)12-s + (−0.800 − 0.800i)13-s − 0.925i·14-s + (−0.190 + 0.708i)15-s + 0.791·16-s + (0.842 − 0.842i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(0.1823529760\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1823529760\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (7.62e10 + 1.32e11i)T \) |
good | 2 | \( 1 + (4.20e4 - 4.20e4i)T - 4.29e9iT^{2} \) |
| 3 | \( 1 + (2.23e7 + 2.23e7i)T + 1.85e15iT^{2} \) |
| 7 | \( 1 + (2.39e13 - 2.39e13i)T - 1.10e27iT^{2} \) |
| 11 | \( 1 + 3.17e16T + 2.11e33T^{2} \) |
| 13 | \( 1 + (5.32e17 + 5.32e17i)T + 4.42e35iT^{2} \) |
| 17 | \( 1 + (-4.10e19 + 4.10e19i)T - 2.36e39iT^{2} \) |
| 19 | \( 1 + 2.22e20iT - 8.31e40T^{2} \) |
| 23 | \( 1 + (6.22e21 + 6.22e21i)T + 3.76e43iT^{2} \) |
| 29 | \( 1 - 3.25e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 - 1.21e22T + 5.29e47T^{2} \) |
| 37 | \( 1 + (1.53e25 - 1.53e25i)T - 1.52e50iT^{2} \) |
| 41 | \( 1 + 2.70e25T + 4.06e51T^{2} \) |
| 43 | \( 1 + (-9.54e25 - 9.54e25i)T + 1.86e52iT^{2} \) |
| 47 | \( 1 + (-2.53e26 + 2.53e26i)T - 3.21e53iT^{2} \) |
| 53 | \( 1 + (1.49e27 + 1.49e27i)T + 1.50e55iT^{2} \) |
| 59 | \( 1 + 2.48e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 3.17e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + (3.16e27 - 3.16e27i)T - 2.71e58iT^{2} \) |
| 71 | \( 1 - 4.69e29T + 1.73e59T^{2} \) |
| 73 | \( 1 + (3.04e29 + 3.04e29i)T + 4.22e59iT^{2} \) |
| 79 | \( 1 - 1.79e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + (4.68e29 + 4.68e29i)T + 2.57e61iT^{2} \) |
| 89 | \( 1 - 2.97e31iT - 2.40e62T^{2} \) |
| 97 | \( 1 + (1.89e31 - 1.89e31i)T - 3.77e63iT^{2} \) |
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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
−16.41367927458907299366571381704, −15.44207762038731909064913205435, −12.67556484855137989660319929890, −12.15933757192349503633969811890, −9.552623557423060161907078090631, −8.216147647652124879331814929009, −6.89401720925537780754810672576, −5.34959075517241860559537467105, −3.08362096922098686678761327556, −0.59642719337839519039370033244,
0.13558411464732156635736030620, 2.13623157535274191655076133761, 3.82860377516859546656039771106, 5.79012678112373894454549817175, 7.65144237621860320529491927046, 9.967404189868347357731598524315, 10.50233943595461738569563738591, 11.85823028968646577902770525857, 14.15363768640244492134143940274, 15.76019016393890681252317818184