| L(s) = 1 | + (1.83 + 1.49i)2-s + (−3.22 − 0.260i)3-s + (0.722 + 3.53i)4-s + (−0.992 + 0.120i)5-s + (−5.53 − 5.31i)6-s + (−2.51 − 3.98i)7-s + (−1.77 + 3.38i)8-s + (7.40 + 1.20i)9-s + (−2.00 − 1.26i)10-s + (0.570 + 3.51i)11-s + (−1.41 − 11.6i)12-s + (1.80 − 3.12i)13-s + (1.34 − 11.0i)14-s + (3.23 − 0.130i)15-s + (−1.67 + 0.715i)16-s + (−0.374 + 0.236i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 1.05i)2-s + (−1.86 − 0.150i)3-s + (0.361 + 1.76i)4-s + (−0.443 + 0.0539i)5-s + (−2.26 − 2.17i)6-s + (−0.952 − 1.50i)7-s + (−0.627 + 1.19i)8-s + (2.46 + 0.401i)9-s + (−0.633 − 0.400i)10-s + (0.172 + 1.05i)11-s + (−0.407 − 3.35i)12-s + (0.500 − 0.865i)13-s + (0.359 − 2.96i)14-s + (0.835 − 0.0336i)15-s + (−0.419 + 0.178i)16-s + (−0.0908 + 0.0574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36423 + 0.410975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36423 + 0.410975i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.992 - 0.120i)T \) |
| 13 | \( 1 + (-1.80 + 3.12i)T \) |
| good | 2 | \( 1 + (-1.83 - 1.49i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (3.22 + 0.260i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (2.51 + 3.98i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (-0.570 - 3.51i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (0.374 - 0.236i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (-4.17 - 2.41i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.164 - 0.285i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.69 + 5.74i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (0.233 - 0.946i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (-2.50 + 0.724i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (-0.352 + 4.36i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (-0.858 + 2.96i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (-0.210 + 0.237i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (-10.6 - 5.58i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (-4.83 + 11.3i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-0.397 + 9.86i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (13.3 + 2.73i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (-1.73 + 0.822i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (-10.3 - 3.92i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (2.65 + 2.35i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-5.02 + 3.46i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (-0.606 + 0.350i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.78 + 5.03i)T + (-26.9 - 93.1i)T^{2} \) |
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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
−10.35704503584265897491227467427, −9.901571764831008447877838059439, −7.74845169171237500907046952353, −7.28004840265236357794506390005, −6.60654191007853836840322541069, −5.97943071745294287899116920030, −5.05182927760448674861678225785, −4.25054617359777245301976053627, −3.58523149630921092349334049540, −0.75649040979685614980296764355,
1.07545001123980477450233624887, 2.79046534295355450673376875586, 3.81794765744554578837169335377, 4.83057304426320826353853708133, 5.59194903367670630293554116965, 6.11898899530613733200207499029, 6.86578079900303171464702998149, 8.770034403686341383221398546860, 9.681644895173049097965704386126, 10.62971933783808467698618293046
