| L(s) = 1 | + (−1.26 + 2.72i)3-s + 0.735·7-s + (−5.82 − 6.86i)9-s + 10.9i·11-s + 21.1·13-s + 7.03i·17-s − 23.1·19-s + (−0.927 + 2.00i)21-s + 24.7i·23-s + (26.0 − 7.21i)27-s + 32.3i·29-s − 34.9·31-s + (−29.7 − 13.7i)33-s − 37.7·37-s + (−26.6 + 57.6i)39-s + ⋯ |
| L(s) = 1 | + (−0.420 + 0.907i)3-s + 0.105·7-s + (−0.647 − 0.762i)9-s + 0.995i·11-s + 1.63·13-s + 0.413i·17-s − 1.21·19-s + (−0.0441 + 0.0953i)21-s + 1.07i·23-s + (0.963 − 0.267i)27-s + 1.11i·29-s − 1.12·31-s + (−0.902 − 0.417i)33-s − 1.02·37-s + (−0.684 + 1.47i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.000076204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.000076204\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.26 - 2.72i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.735T + 49T^{2} \) |
| 11 | \( 1 - 10.9iT - 121T^{2} \) |
| 13 | \( 1 - 21.1T + 169T^{2} \) |
| 17 | \( 1 - 7.03iT - 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 - 24.7iT - 529T^{2} \) |
| 29 | \( 1 - 32.3iT - 841T^{2} \) |
| 31 | \( 1 + 34.9T + 961T^{2} \) |
| 37 | \( 1 + 37.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 22.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 39.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 60.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 7.79iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 11.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 96.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 121.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
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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
−10.81329377975059632009510522679, −10.11203144696887871357875103730, −9.075988130309232248260723408374, −8.503924467342673064245854006549, −7.16353633391497245051453200127, −6.16333952569820813852357334606, −5.30829984826742790847580055902, −4.20822379946978209462762793611, −3.44984525654220185990875028214, −1.64888375415577776367758233248,
0.39834147643748665494144002692, 1.74372092870423594006632996297, 3.14054665788724809332783204402, 4.47783298796951089684377720237, 5.90447520352406455641773284999, 6.23690616246403720024125058575, 7.36645151128226672511625997531, 8.449028197268347050353677235064, 8.787619345595090451056486996234, 10.41952341864439013449935239994
