| L(s) = 1 | + (−1.51 + 0.845i)3-s + (−3.37 − 1.22i)5-s + (−0.462 + 2.62i)7-s + (1.57 − 2.55i)9-s + (−0.236 + 0.0860i)11-s + (4.63 − 3.88i)13-s + (6.14 − 0.996i)15-s + (2.21 + 3.84i)17-s + (3.90 − 6.76i)19-s + (−1.51 − 4.35i)21-s + (−0.688 − 3.90i)23-s + (6.05 + 5.08i)25-s + (−0.214 + 5.19i)27-s + (−3.97 − 3.33i)29-s + (0.0109 + 0.0622i)31-s + ⋯ |
| L(s) = 1 | + (−0.872 + 0.488i)3-s + (−1.50 − 0.549i)5-s + (−0.174 + 0.991i)7-s + (0.523 − 0.851i)9-s + (−0.0713 + 0.0259i)11-s + (1.28 − 1.07i)13-s + (1.58 − 0.257i)15-s + (0.537 + 0.931i)17-s + (0.895 − 1.55i)19-s + (−0.331 − 0.950i)21-s + (−0.143 − 0.814i)23-s + (1.21 + 1.01i)25-s + (−0.0412 + 0.999i)27-s + (−0.738 − 0.619i)29-s + (0.00197 + 0.0111i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.682481 - 0.219767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.682481 - 0.219767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.51 - 0.845i)T \) |
| good | 5 | \( 1 + (3.37 + 1.22i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.462 - 2.62i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.236 - 0.0860i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-4.63 + 3.88i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 3.84i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.90 + 6.76i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.688 + 3.90i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.97 + 3.33i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.0109 - 0.0622i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.23 - 2.13i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.28 + 5.27i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.30 - 0.838i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.795 + 4.51i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.892T + 53T^{2} \) |
| 59 | \( 1 + (-4.57 - 1.66i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.928 - 5.26i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 + 1.92i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.66 + 6.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.20 + 5.55i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.99 + 7.54i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.891 - 0.747i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-4.40 + 7.62i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.78 - 0.651i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.22026548215503941053645547247, −10.40492830722660110159597965448, −9.111575181381092337593658137856, −8.437538167053789275604636224671, −7.44245686068789080290493502877, −6.10401327401914149063864698172, −5.30982104097088204587699259050, −4.21765483231213385016508148358, −3.25161886148308274901624087089, −0.64637312851818537990947899287,
1.16190809360312477751693551066, 3.50925359941817043150474848286, 4.19347775751036777488370376147, 5.64064839708225308534025218935, 6.80673162597378859218933903428, 7.44219706263696983956201340557, 8.071309567977668807496309512944, 9.633150427047007846591214928925, 10.69569191810739105286410905745, 11.39911562500181793432239948359
