| L(s) = 1 | + (1.06 + 0.929i)2-s + (0.533 − 1.64i)3-s + (0.272 + 1.98i)4-s + (2.76 + 2.05i)5-s + (2.09 − 1.26i)6-s + (2.24 − 3.41i)7-s + (−1.55 + 2.36i)8-s + (−2.43 − 1.75i)9-s + (1.03 + 4.75i)10-s + (−6.08 + 0.711i)11-s + (3.41 + 0.606i)12-s + (−0.110 + 0.368i)13-s + (5.56 − 1.55i)14-s + (4.85 − 3.45i)15-s + (−3.85 + 1.08i)16-s + (−0.0545 + 0.00961i)17-s + ⋯ |
| L(s) = 1 | + (0.753 + 0.657i)2-s + (0.307 − 0.951i)3-s + (0.136 + 0.990i)4-s + (1.23 + 0.919i)5-s + (0.857 − 0.514i)6-s + (0.847 − 1.28i)7-s + (−0.548 + 0.836i)8-s + (−0.810 − 0.585i)9-s + (0.326 + 1.50i)10-s + (−1.83 + 0.214i)11-s + (0.984 + 0.175i)12-s + (−0.0305 + 0.102i)13-s + (1.48 − 0.414i)14-s + (1.25 − 0.891i)15-s + (−0.962 + 0.270i)16-s + (−0.0132 + 0.00233i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.35300 + 0.522177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35300 + 0.522177i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 - 0.929i)T \) |
| 3 | \( 1 + (-0.533 + 1.64i)T \) |
| good | 5 | \( 1 + (-2.76 - 2.05i)T + (1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-2.24 + 3.41i)T + (-2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (6.08 - 0.711i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.110 - 0.368i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.0545 - 0.00961i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-2.12 - 0.373i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.778 - 0.511i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (1.01 + 0.953i)T + (1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-8.84 + 0.515i)T + (30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (7.21 + 2.62i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-1.17 - 4.94i)T + (-36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (8.50 + 3.66i)T + (29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (0.106 - 1.82i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-0.0530 + 0.0306i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.72 + 0.318i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-8.15 + 4.09i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (6.31 - 5.95i)T + (3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (4.63 - 3.88i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.203 + 0.171i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.74 + 7.37i)T + (-70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (-3.01 - 0.713i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (-8.55 + 10.1i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (5.36 + 7.20i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−11.84510789924153925187477642830, −10.80006200075748685333579512294, −9.994821688921720421584920776622, −8.330127530557010614897306374181, −7.56086693309571242625908516471, −6.93289700859284206927568506249, −5.90381143379055207478981102798, −4.89693758692651984471846282538, −3.15926170071048327969844793392, −2.08453072961653154887031111101,
2.03079171165510458294151761670, 2.91619010640592166354236044228, 4.88568764282991217062380916097, 5.17535218856047050709562113171, 5.90679224746208031228590491732, 8.219818463475420034745047966396, 8.936715781542031245697274915746, 9.907613241719902456705154326246, 10.49870978774229520116723146984, 11.59160556024448833805071727670
