| 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
−13.43328773236877304004880631744, −12.41729474109219932409144843832, −11.26399645911647329017060567023, −10.36440987977612283933737326605, −8.557428479630086110325286869648, −8.264233353990898729008714900055, −6.68146448524100227242404020360, −5.83411941093625982145067564158, −4.55646159590577678250025061701, −2.06107111598418183696851393998,
1.60348294142352999779041223287, 3.95403265836380024578165053345, 4.76973497030798439490305308654, 6.34746652084959767542173891480, 8.175595574823469265691906417456, 8.893878714161657077552350837240, 10.25220130920474447014060194275, 11.09866957460464632849491595057, 11.59861340891831015000962308433, 13.14924724539727027220956362458
