| L(s) = 1 | + (−0.988 − 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (−0.292 − 3.90i)5-s + (0.900 − 0.433i)6-s + (−2.59 + 0.494i)7-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (−0.292 + 3.90i)10-s + (1.49 + 3.80i)11-s + (−0.955 + 0.294i)12-s + (−4.20 + 5.27i)13-s + (2.64 − 0.101i)14-s + (2.44 + 3.06i)15-s + (0.826 + 0.563i)16-s + (−4.10 − 3.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (−0.477 + 0.325i)3-s + (0.477 + 0.147i)4-s + (−0.130 − 1.74i)5-s + (0.367 − 0.177i)6-s + (−0.982 + 0.186i)7-s + (−0.318 − 0.153i)8-s + (0.121 − 0.310i)9-s + (−0.0926 + 1.23i)10-s + (0.450 + 1.14i)11-s + (−0.275 + 0.0850i)12-s + (−1.16 + 1.46i)13-s + (0.706 − 0.0272i)14-s + (0.630 + 0.791i)15-s + (0.206 + 0.140i)16-s + (−0.996 − 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00619500 + 0.0614870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00619500 + 0.0614870i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 3 | \( 1 + (0.826 - 0.563i)T \) |
| 7 | \( 1 + (2.59 - 0.494i)T \) |
| good | 5 | \( 1 + (0.292 + 3.90i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.49 - 3.80i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (4.20 - 5.27i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.10 + 3.81i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (2.25 + 3.90i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 2.40i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.826 + 3.61i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.37 - 4.11i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.386 + 0.119i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (2.99 + 1.44i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-2.08 + 1.00i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.59 - 0.390i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (6.76 + 2.08i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.0803 + 1.07i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (6.65 - 2.05i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.550 - 0.953i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.17 + 9.51i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-9.93 + 1.49i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-4.87 - 8.43i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.58 + 10.7i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.54 + 3.94i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−11.48381586187358630035788642903, −9.935962309266855999441209324066, −9.259709716817972143047507643647, −8.985498671047975862177466700839, −7.34884494345511189514234510577, −6.49856470480762996072375991228, −4.94603754918739007482887810989, −4.27545421411336292516293791240, −2.03864049088153415372272304082, −0.05599184159425724752137073492,
2.54337578249580738746483384740, 3.61285720836088607401323901858, 5.95235281286075702727396940748, 6.41628994671291136135008208518, 7.35619625359669410945643857276, 8.263570387258954993234163223919, 9.746643190277637285320122979666, 10.59733616824121777830049801637, 10.88159890324629943314348898469, 12.11769241931738447387438829379
