| L(s) = 1 | + (−0.642 + 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 − 0.984i)4-s + (−3.23 − 0.570i)5-s + (−0.984 + 0.173i)6-s + (0.262 − 2.63i)7-s + (0.866 + 0.500i)8-s + (0.173 + 0.984i)9-s + (2.51 − 2.11i)10-s + (0.0638 + 0.110i)11-s + (0.500 − 0.866i)12-s + (0.870 + 4.93i)13-s + (1.84 + 1.89i)14-s + (−2.11 − 2.51i)15-s + (−0.939 + 0.342i)16-s + (−3.13 − 0.552i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.442 + 0.371i)3-s + (−0.0868 − 0.492i)4-s + (−1.44 − 0.255i)5-s + (−0.402 + 0.0708i)6-s + (0.0993 − 0.995i)7-s + (0.306 + 0.176i)8-s + (0.0578 + 0.328i)9-s + (0.796 − 0.668i)10-s + (0.0192 + 0.0333i)11-s + (0.144 − 0.249i)12-s + (0.241 + 1.36i)13-s + (0.493 + 0.506i)14-s + (−0.545 − 0.650i)15-s + (−0.234 + 0.0855i)16-s + (−0.759 − 0.133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.736941 + 0.660914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.736941 + 0.660914i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.262 + 2.63i)T \) |
| 19 | \( 1 + (-3.96 - 1.82i)T \) |
| good | 5 | \( 1 + (3.23 + 0.570i)T + (4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.0638 - 0.110i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.870 - 4.93i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.13 + 0.552i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.30 - 1.93i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.809 - 2.22i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + (0.581 - 0.335i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.94 - 11.0i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-9.67 - 8.12i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.29 - 0.228i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.39 + 1.30i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.153 + 0.869i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.946 - 2.60i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (10.3 + 12.3i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 1.83i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (6.11 - 7.28i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (4.28 + 11.7i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.9 - 6.29i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.84 + 6.58i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (7.06 - 2.57i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.35370359238633500461464312695, −9.410713330750841980633040856360, −8.705181595777952412040817524327, −7.83931236105062783933539787480, −7.29685620785854662017686166349, −6.43488581786908493767589519572, −4.69923293705943255332909197615, −4.33329778582959130301784075496, −3.20417391769827365447207509660, −1.16731464886033594502443039190,
0.67121231665473042621844050744, 2.55300616511881395094456180886, 3.22135138489900349353032920454, 4.34840513044367782976107673362, 5.62300581390018114446309675316, 7.00308238905890248130613851048, 7.66522528439519003307956038147, 8.554021993108623154806019389030, 8.866113715582067716779042845915, 10.16656319001211396127517055637
