| L(s) = 1 | + (−11.3 − 11.2i)2-s + (1.63 + 255. i)4-s + 279.·5-s − 3.94e3i·7-s + (2.86e3 − 2.92e3i)8-s + (−3.17e3 − 3.15e3i)10-s + 1.70e4i·11-s + 1.09e4·13-s + (−4.44e4 + 4.47e4i)14-s + (−6.55e4 + 837. i)16-s − 1.01e5·17-s + 9.34e4i·19-s + (457. + 7.15e4i)20-s + (1.92e5 − 1.94e5i)22-s − 1.47e5i·23-s + ⋯ |
| L(s) = 1 | + (−0.709 − 0.704i)2-s + (0.00639 + 0.999i)4-s + 0.447·5-s − 1.64i·7-s + (0.700 − 0.713i)8-s + (−0.317 − 0.315i)10-s + 1.16i·11-s + 0.382·13-s + (−1.15 + 1.16i)14-s + (−0.999 + 0.0127i)16-s − 1.21·17-s + 0.716i·19-s + (0.00285 + 0.447i)20-s + (0.823 − 0.828i)22-s − 0.526i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00639 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.00639 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3341353300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3341353300\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.3 + 11.2i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 279.T \) |
| good | 7 | \( 1 + 3.94e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.70e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.09e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.01e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 9.34e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.47e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.16e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.38e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.14e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.83e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 3.18e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.66e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.69e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 5.16e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.05e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.85e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.38e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.42e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 1.51e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.42e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 1.24e8T + 7.83e15T^{2} \) |
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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
−11.10210863186753393544973083086, −10.35460226751896879916480858536, −9.721901508882891966224189437313, −8.517933511557002019733042748619, −7.38173323982006823345870805861, −6.64686125812723754121617609752, −4.59595325198290701200134984473, −3.75466591114392610434391364223, −2.18032572012826401310205747038, −1.15605057795735388167475883596,
0.10438401437496843259436218488, 1.65267599237229558584992408841, 2.82719348666686400889878285574, 4.93830521531777281282504151724, 5.90347288770767790162595891953, 6.55653798032813083003453036286, 8.148154879694937234800532247318, 8.888126556940968501772975696849, 9.485589092007435790418357303506, 10.91646816622167295688837457893
