| L(s) = 1 | + (−1.40 + 0.119i)2-s + (−1.73 + 0.0550i)3-s + (1.97 − 0.337i)4-s + (2.63 − 1.20i)5-s + (2.43 − 0.284i)6-s + (−0.694 − 0.728i)7-s + (−2.73 + 0.711i)8-s + (2.99 − 0.190i)9-s + (−3.56 + 2.00i)10-s + (1.29 + 0.123i)11-s + (−3.39 + 0.692i)12-s + (5.25 + 1.01i)13-s + (1.06 + 0.943i)14-s + (−4.48 + 2.22i)15-s + (3.77 − 1.32i)16-s + (−2.41 + 4.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0846i)2-s + (−0.999 + 0.0317i)3-s + (0.985 − 0.168i)4-s + (1.17 − 0.537i)5-s + (0.993 − 0.116i)6-s + (−0.262 − 0.275i)7-s + (−0.967 + 0.251i)8-s + (0.997 − 0.0634i)9-s + (−1.12 + 0.635i)10-s + (0.390 + 0.0372i)11-s + (−0.979 + 0.199i)12-s + (1.45 + 0.280i)13-s + (0.284 + 0.252i)14-s + (−1.15 + 0.574i)15-s + (0.943 − 0.332i)16-s + (−0.585 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.939128 - 0.149745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939128 - 0.149745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.119i)T \) |
| 3 | \( 1 + (1.73 - 0.0550i)T \) |
| 67 | \( 1 + (-7.61 - 3.00i)T \) |
| good | 5 | \( 1 + (-2.63 + 1.20i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (0.694 + 0.728i)T + (-0.333 + 6.99i)T^{2} \) |
| 11 | \( 1 + (-1.29 - 0.123i)T + (10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-5.25 - 1.01i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (2.41 - 4.68i)T + (-9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (-3.97 + 4.16i)T + (-0.904 - 18.9i)T^{2} \) |
| 23 | \( 1 + (2.60 - 1.04i)T + (16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (0.207 + 0.120i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.254 + 1.32i)T + (-28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.96 - 0.141i)T + (40.8 - 3.89i)T^{2} \) |
| 43 | \( 1 + (-6.47 - 10.0i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (-7.55 - 5.94i)T + (11.0 + 45.6i)T^{2} \) |
| 53 | \( 1 + (-5.23 + 8.14i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (6.62 + 7.64i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (4.91 - 0.468i)T + (59.8 - 11.5i)T^{2} \) |
| 71 | \( 1 + (0.00404 - 0.00208i)T + (41.1 - 57.8i)T^{2} \) |
| 73 | \( 1 + (-7.73 + 0.739i)T + (71.6 - 13.8i)T^{2} \) |
| 79 | \( 1 + (-13.1 + 4.54i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (0.00396 - 0.00556i)T + (-27.1 - 78.4i)T^{2} \) |
| 89 | \( 1 + (-11.8 + 1.69i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (2.92 + 5.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.14296778618385485819553094838, −9.404495396188964168148052024132, −8.825325792532717934140916195689, −7.63724945097864513886802769892, −6.40851749393759847294667487167, −6.21719419263325461622906447546, −5.20715656471432415680002891886, −3.79774091594774221376712085871, −1.91978421921738772772245145603, −0.987293344334771401453592307226,
1.08836648820867833386529542480, 2.29308364156950276912445677784, 3.66757117222325543398066339101, 5.48070442638202984704341497530, 6.07606372137197408737903393195, 6.70356734590464947935628632845, 7.62832473469935700314689034591, 8.930643731234243262107495834492, 9.520227940989263582638565076613, 10.44757800360918406403559121497
