| L(s) = 1 | + (0.119 − 1.36i)2-s + (−1.50 + 0.861i)3-s + (0.116 + 0.0204i)4-s + (1.52 + 1.63i)5-s + (0.997 + 2.15i)6-s + (1.87 − 2.68i)7-s + (0.752 − 2.80i)8-s + (1.51 − 2.58i)9-s + (2.41 − 1.89i)10-s + (−1.85 + 5.10i)11-s + (−0.192 + 0.0692i)12-s + (2.45 − 0.215i)13-s + (−3.44 − 2.88i)14-s + (−3.70 − 1.14i)15-s + (−3.52 − 1.28i)16-s + (−0.925 − 3.45i)17-s + ⋯ |
| L(s) = 1 | + (0.0845 − 0.966i)2-s + (−0.867 + 0.497i)3-s + (0.0580 + 0.0102i)4-s + (0.682 + 0.730i)5-s + (0.407 + 0.880i)6-s + (0.710 − 1.01i)7-s + (0.265 − 0.992i)8-s + (0.505 − 0.862i)9-s + (0.763 − 0.598i)10-s + (−0.560 + 1.53i)11-s + (−0.0554 + 0.0199i)12-s + (0.682 − 0.0596i)13-s + (−0.920 − 0.772i)14-s + (−0.955 − 0.294i)15-s + (−0.881 − 0.320i)16-s + (−0.224 − 0.837i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04771 - 0.406864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04771 - 0.406864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.50 - 0.861i)T \) |
| 5 | \( 1 + (-1.52 - 1.63i)T \) |
| good | 2 | \( 1 + (-0.119 + 1.36i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.87 + 2.68i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.85 - 5.10i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.45 + 0.215i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (0.925 + 3.45i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.417 - 0.240i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.73 - 4.01i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (0.0993 - 0.0833i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.509 + 2.88i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (6.26 - 1.67i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.215 + 0.256i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.27 - 2.45i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (9.71 + 6.80i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-0.167 - 0.167i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.62 + 1.32i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.855 + 4.85i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.372 - 4.25i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-7.10 - 4.10i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.1 - 3.51i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.44 + 7.68i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.0147 + 0.00128i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (-2.55 - 4.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 3.77i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−12.93313004399237615664885107767, −11.72660828128703869984746003665, −11.08483185474398844823389011329, −10.14128000625369896982741022355, −9.826667215827025108312143911155, −7.45706163239267369837225430766, −6.58875653482838226609635025786, −4.97263218794373936884516624988, −3.72495617564892417011763973843, −1.82664702424434893379536437394,
1.93638131442699183004109298998, 5.01011329503178758466910555412, 5.82065955578901211822209326201, 6.37116961859371542434939516178, 8.211343162108156294119838214596, 8.518874071027383485989676730512, 10.53645203178505371108899888298, 11.37630947836385932600137224675, 12.37709730007453352093215022825, 13.47191749996842795997022236134
