| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (0.263 − 1.49i)5-s + (0.173 + 0.984i)6-s + (−1.92 + 1.81i)7-s + (−0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (1.16 − 0.973i)10-s + (1.75 + 3.04i)11-s + (−0.500 + 0.866i)12-s + (0.432 + 2.45i)13-s + (−2.64 + 0.158i)14-s + (1.16 − 0.973i)15-s + (−0.939 + 0.342i)16-s + (−0.696 + 3.95i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.442 + 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.117 − 0.667i)5-s + (0.0708 + 0.402i)6-s + (−0.726 + 0.687i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.366 − 0.307i)10-s + (0.529 + 0.916i)11-s + (−0.144 + 0.249i)12-s + (0.120 + 0.680i)13-s + (−0.705 + 0.0423i)14-s + (0.299 − 0.251i)15-s + (−0.234 + 0.0855i)16-s + (−0.168 + 0.958i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31409 + 1.73899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31409 + 1.73899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (1.92 - 1.81i)T \) |
| 19 | \( 1 + (2.42 + 3.62i)T \) |
| good | 5 | \( 1 + (-0.263 + 1.49i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 3.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.432 - 2.45i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.696 - 3.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.412 - 0.150i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-3.36 - 1.22i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 + (0.755 + 1.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.80 - 10.2i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.09 - 3.43i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.85 + 10.5i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.774 - 4.39i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.58 + 8.97i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (9.68 + 3.52i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-7.59 + 6.37i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (10.5 + 8.86i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.20 - 1.85i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-12.3 + 4.50i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (5.14 + 8.90i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.81 - 4.04i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.99 + 2.90i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.37173621975025800137414039887, −9.353965012829791454244169977982, −8.927629918543207371178784808513, −8.082975547325466886449456991265, −6.79569500824815537438956993696, −6.25454970114588277416225312889, −4.95491849065291962430545773650, −4.36365487255434279761362988543, −3.20381220299476189124226735857, −1.95278340539008909321579404489,
0.895386861898907731343306312216, 2.62474093703771677816300686829, 3.32016114488987555012634575158, 4.25195563496366349508211205016, 5.73577430407886810431615348290, 6.51696070032818367453341707085, 7.22196132217578133071116667094, 8.345010445467633435973239230045, 9.291536081830140510996701003871, 10.27614287131366991264891222780
