| L(s) = 1 | + (0.460 − 1.71i)2-s + (−1.57 − 2.72i)3-s + (0.726 + 0.419i)4-s + (−5.39 + 1.44i)6-s + (−1.30 − 4.87i)7-s + (6.08 − 6.08i)8-s + (−0.437 + 0.758i)9-s + (−3.62 − 0.970i)11-s − 2.63i·12-s + (12.9 + 0.340i)13-s − 8.97·14-s + (−5.97 − 10.3i)16-s + (−23.0 − 13.2i)17-s + (1.10 + 1.10i)18-s + (−15.0 + 4.03i)19-s + ⋯ |
| L(s) = 1 | + (0.230 − 0.858i)2-s + (−0.523 − 0.907i)3-s + (0.181 + 0.104i)4-s + (−0.899 + 0.241i)6-s + (−0.186 − 0.696i)7-s + (0.760 − 0.760i)8-s + (−0.0486 + 0.0842i)9-s + (−0.329 − 0.0882i)11-s − 0.219i·12-s + (0.999 + 0.0261i)13-s − 0.641·14-s + (−0.373 − 0.646i)16-s + (−1.35 − 0.781i)17-s + (0.0611 + 0.0611i)18-s + (−0.792 + 0.212i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00953289 - 1.53065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00953289 - 1.53065i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (-12.9 - 0.340i)T \) |
| good | 2 | \( 1 + (-0.460 + 1.71i)T + (-3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (1.57 + 2.72i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (1.30 + 4.87i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (3.62 + 0.970i)T + (104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (23.0 + 13.2i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (15.0 - 4.03i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-6.64 + 3.83i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-7.79 - 13.4i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-7.36 - 7.36i)T + 961iT^{2} \) |
| 37 | \( 1 + (0.251 + 0.0673i)T + (1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-5.39 + 20.1i)T + (-1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-6.34 - 3.66i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (57.7 - 57.7i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 25.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (14.8 + 55.5i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (27.1 - 47.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-19.7 + 73.6i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-37.8 + 10.1i)T + (4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-71.8 + 71.8i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 123.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (82.4 + 82.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-93.8 - 25.1i)T + (6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-126. + 34.0i)T + (8.14e3 - 4.70e3i)T^{2} \) |
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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
−11.03173996626409400422737005587, −10.51712501146855761875351880875, −9.183942433481456057513490244405, −7.86433000128676310682710104080, −6.86229353914918656154505971892, −6.33234519834677066677743977451, −4.59711709138659031181664865767, −3.43958949572265285047166001661, −2.00401399132982104465463538238, −0.70275344119415235925784993024,
2.18520261378423721115420773551, 4.08971206882045419717255680226, 5.05638988960709519772429206471, 5.98089738608566581489030826030, 6.67411957682708147150580025185, 8.076504570642203838546525114091, 8.910585028468334388615277879165, 10.18347018215403571115191356017, 10.91795897858481522440949414911, 11.51141621107243889837080973713
