| L(s) = 1 | + (0.342 + 0.939i)2-s + (0.939 − 0.342i)3-s + (−0.766 + 0.642i)4-s + (1.12 − 1.33i)5-s + (0.642 + 0.766i)6-s + (2.07 − 1.64i)7-s + (−0.866 − 0.500i)8-s + (0.766 − 0.642i)9-s + (1.63 + 0.596i)10-s + (−2.08 − 3.61i)11-s + (−0.500 + 0.866i)12-s + (−1.65 + 1.38i)13-s + (2.25 + 1.38i)14-s + (0.596 − 1.63i)15-s + (0.173 − 0.984i)16-s + (4.92 − 5.86i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (0.542 − 0.197i)3-s + (−0.383 + 0.321i)4-s + (0.501 − 0.597i)5-s + (0.262 + 0.312i)6-s + (0.784 − 0.620i)7-s + (−0.306 − 0.176i)8-s + (0.255 − 0.214i)9-s + (0.518 + 0.188i)10-s + (−0.630 − 1.09i)11-s + (−0.144 + 0.249i)12-s + (−0.458 + 0.384i)13-s + (0.601 + 0.371i)14-s + (0.154 − 0.423i)15-s + (0.0434 − 0.246i)16-s + (1.19 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.20581 - 0.350369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20581 - 0.350369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-2.07 + 1.64i)T \) |
| 19 | \( 1 + (3.22 + 2.93i)T \) |
| good | 5 | \( 1 + (-1.12 + 1.33i)T + (-0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (2.08 + 3.61i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.65 - 1.38i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.92 + 5.86i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.03 - 5.87i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.14 + 0.202i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 + (-0.370 + 0.213i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.91 + 4.96i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.89 - 1.05i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.61 - 7.88i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-8.08 - 9.63i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.90 - 5.79i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-8.50 + 1.50i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.21 + 3.34i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.09 - 11.2i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.17 - 11.4i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (10.5 + 1.85i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-13.2 + 7.65i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (14.7 + 5.36i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.13 - 17.7i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.997230726117512194209328288227, −9.149310381169399336164583283376, −8.468323201247804205976939774999, −7.56055883718737511623415324243, −7.03233550024431734757304261524, −5.54671123221343102997262323774, −5.11648314881339769754922284558, −3.90645147651759305709108350722, −2.67897081049706263739424739247, −1.04940906881141137368682181134,
1.88916774441711082753747320615, 2.49116337240989348375473267899, 3.74812453166062002189980049207, 4.84826752018626026426560333240, 5.66208870941191458061345319898, 6.82285722772074323085810620997, 8.100276883102925585643179522819, 8.497022225493427072321214683816, 9.920995379958527386657406244291, 10.19023909765956747557815721196
