| L(s) = 1 | + (−0.261 + 0.0124i)2-s + (0.845 + 1.51i)3-s + (−1.92 + 0.183i)4-s + (−3.04 − 2.90i)5-s + (−0.239 − 0.384i)6-s + (−0.702 − 3.64i)7-s + (1.01 − 0.146i)8-s + (−1.57 + 2.55i)9-s + (0.832 + 0.721i)10-s + (−3.45 + 1.78i)11-s + (−1.90 − 2.75i)12-s + (0.259 + 0.0499i)13-s + (0.229 + 0.944i)14-s + (1.81 − 7.05i)15-s + (3.52 − 0.680i)16-s + (−1.22 − 2.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.184 + 0.00881i)2-s + (0.488 + 0.872i)3-s + (−0.961 + 0.0917i)4-s + (−1.36 − 1.29i)5-s + (−0.0979 − 0.157i)6-s + (−0.265 − 1.37i)7-s + (0.360 − 0.0518i)8-s + (−0.523 + 0.851i)9-s + (0.263 + 0.228i)10-s + (−1.04 + 0.537i)11-s + (−0.549 − 0.794i)12-s + (0.0718 + 0.0138i)13-s + (0.0612 + 0.252i)14-s + (0.468 − 1.82i)15-s + (0.882 − 0.170i)16-s + (−0.295 − 0.647i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.143131 - 0.299936i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143131 - 0.299936i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.845 - 1.51i)T \) |
| 23 | \( 1 + (3.98 + 2.67i)T \) |
| good | 2 | \( 1 + (0.261 - 0.0124i)T + (1.99 - 0.190i)T^{2} \) |
| 5 | \( 1 + (3.04 + 2.90i)T + (0.237 + 4.99i)T^{2} \) |
| 7 | \( 1 + (0.702 + 3.64i)T + (-6.49 + 2.60i)T^{2} \) |
| 11 | \( 1 + (3.45 - 1.78i)T + (6.38 - 8.96i)T^{2} \) |
| 13 | \( 1 + (-0.259 - 0.0499i)T + (12.0 + 4.83i)T^{2} \) |
| 17 | \( 1 + (1.22 + 2.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.48 - 0.679i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.234 + 2.45i)T + (-28.4 - 5.48i)T^{2} \) |
| 31 | \( 1 + (-0.601 + 0.472i)T + (7.30 - 30.1i)T^{2} \) |
| 37 | \( 1 + (-1.05 + 3.58i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-2.75 + 2.89i)T + (-1.95 - 40.9i)T^{2} \) |
| 43 | \( 1 + (4.40 - 5.60i)T + (-10.1 - 41.7i)T^{2} \) |
| 47 | \( 1 + (4.75 + 2.74i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.42 - 1.64i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.82 - 9.49i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-1.51 + 3.79i)T + (-44.1 - 42.0i)T^{2} \) |
| 67 | \( 1 + (-6.13 + 11.9i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (0.327 + 0.510i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.89 + 15.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.99 + 0.691i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (-8.75 + 8.34i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-1.15 + 8.01i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (5.87 + 1.42i)T + (86.2 + 44.4i)T^{2} \) |
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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
−12.15961230225797446187940772589, −10.84546821276619834145224463629, −9.923721813700383523855778958840, −9.086401073004519629940756454786, −7.997974383476283393536753038367, −7.65068622402773788247878417663, −5.02178161665859277261293103305, −4.41891662844710509496552966089, −3.62616834248199996234768071155, −0.28755926499326395076583107792,
2.68914608336531739739076073910, 3.69156319260750152344377013421, 5.58153597488860946523396964164, 6.78659342864008557133218399924, 8.156696632224758960619410553738, 8.266377628602115181898566951652, 9.666259036068957462992525891953, 10.93796654148701387814047133192, 11.88556986519717414977633623029, 12.70326536981961345593960897154
