| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.698 − 0.806i)5-s + (0.959 + 0.281i)6-s + (3.72 − 2.39i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.897 − 0.577i)10-s + (0.234 − 1.63i)11-s + (0.142 − 0.989i)12-s + (4.60 + 2.95i)13-s + (−2.89 − 3.34i)14-s + (0.443 + 0.970i)15-s + (0.841 − 0.540i)16-s + (−3.76 − 1.10i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.312 − 0.360i)5-s + (0.391 + 0.115i)6-s + (1.40 − 0.904i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.283 − 0.182i)10-s + (0.0707 − 0.492i)11-s + (0.0410 − 0.285i)12-s + (1.27 + 0.820i)13-s + (−0.774 − 0.893i)14-s + (0.114 + 0.250i)15-s + (0.210 − 0.135i)16-s + (−0.913 − 0.268i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.982358 - 0.430531i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982358 - 0.430531i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (4.78 - 0.323i)T \) |
| good | 5 | \( 1 + (-0.698 + 0.806i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-3.72 + 2.39i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.234 + 1.63i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-4.60 - 2.95i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (3.76 + 1.10i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (6.33 - 1.85i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-6.14 - 1.80i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (1.25 + 2.74i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 1.67i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (2.67 - 3.09i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.21 - 2.64i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 9.62T + 47T^{2} \) |
| 53 | \( 1 + (1.87 - 1.20i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.20 - 5.27i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.647 + 1.41i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (1.71 + 11.9i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.749 - 5.21i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (6.89 - 2.02i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-2.26 - 1.45i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (3.62 + 4.18i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.70 + 10.3i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (7.84 - 9.05i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−13.14924724539727027220956362458, −11.59861340891831015000962308433, −11.09866957460464632849491595057, −10.25220130920474447014060194275, −8.893878714161657077552350837240, −8.175595574823469265691906417456, −6.34746652084959767542173891480, −4.76973497030798439490305308654, −3.95403265836380024578165053345, −1.60348294142352999779041223287,
2.06107111598418183696851393998, 4.55646159590577678250025061701, 5.83411941093625982145067564158, 6.68146448524100227242404020360, 8.264233353990898729008714900055, 8.557428479630086110325286869648, 10.36440987977612283933737326605, 11.26399645911647329017060567023, 12.41729474109219932409144843832, 13.43328773236877304004880631744
