| L(s) = 1 | + (−1.97 + 0.342i)2-s + (3.76 − 1.34i)4-s − 11.5·7-s + (−6.95 + 3.94i)8-s − 9.97i·11-s + 14.1i·13-s + (22.6 − 3.94i)14-s + (12.3 − 10.1i)16-s − 30.5i·17-s − 12.2i·19-s + (3.41 + 19.6i)22-s + 15.7·23-s + (−4.83 − 27.8i)26-s + (−43.3 + 15.5i)28-s − 18.8·29-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.171i)2-s + (0.941 − 0.337i)4-s − 1.64·7-s + (−0.869 + 0.493i)8-s − 0.906i·11-s + 1.08i·13-s + (1.62 − 0.281i)14-s + (0.772 − 0.635i)16-s − 1.79i·17-s − 0.643i·19-s + (0.155 + 0.893i)22-s + 0.685·23-s + (−0.185 − 1.07i)26-s + (−1.54 + 0.554i)28-s − 0.651·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5305654200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5305654200\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.97 - 0.342i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 11.5T + 49T^{2} \) |
| 11 | \( 1 + 9.97iT - 121T^{2} \) |
| 13 | \( 1 - 14.1iT - 169T^{2} \) |
| 17 | \( 1 + 30.5iT - 289T^{2} \) |
| 19 | \( 1 + 12.2iT - 361T^{2} \) |
| 23 | \( 1 - 15.7T + 529T^{2} \) |
| 29 | \( 1 + 18.8T + 841T^{2} \) |
| 31 | \( 1 - 35.2iT - 961T^{2} \) |
| 37 | \( 1 - 50.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.46iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 64.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 30.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 66.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 11.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 2.24iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 146.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 87.4T + 7.92e3T^{2} \) |
| 97 | \( 1 - 126. iT - 9.40e3T^{2} \) |
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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
−9.805442229996266483349064076323, −9.247153985467021382823749258925, −8.779418475385344677761250400861, −7.43148216804547221892505543529, −6.77869962786883004510009162755, −6.18600094542040749659868803560, −5.00012505410129670201872117072, −3.33744648566827104291306532412, −2.64504849259091715207058840054, −0.879694144404883487115013433888,
0.31009917257067474484768750764, 1.91038813753261758842919008908, 3.11867465476140895873148619153, 3.93209697990197164090860900917, 5.78867924474017633790058415018, 6.32682540589764224261478017888, 7.35499623193906176061850538952, 8.021702493113531850718463139898, 9.111731488980397634558645811196, 9.715224671850525338556451701660
