| L(s) = 1 | − 2.23i·5-s + 9.55i·7-s + 18.8i·11-s + 20.1·13-s − 28.4i·17-s − 28.9i·19-s + (−18.3 + 13.8i)23-s − 5.00·25-s + 1.59·29-s − 26.9·31-s + 21.3·35-s − 51.7i·37-s − 54.4·41-s − 81.2i·43-s + 27.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.36i·7-s + 1.70i·11-s + 1.54·13-s − 1.67i·17-s − 1.52i·19-s + (−0.799 + 0.600i)23-s − 0.200·25-s + 0.0548·29-s − 0.867·31-s + 0.610·35-s − 1.39i·37-s − 1.32·41-s − 1.89i·43-s + 0.586·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.788734081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788734081\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (18.3 - 13.8i)T \) |
| good | 7 | \( 1 - 9.55iT - 49T^{2} \) |
| 11 | \( 1 - 18.8iT - 121T^{2} \) |
| 13 | \( 1 - 20.1T + 169T^{2} \) |
| 17 | \( 1 + 28.4iT - 289T^{2} \) |
| 19 | \( 1 + 28.9iT - 361T^{2} \) |
| 29 | \( 1 - 1.59T + 841T^{2} \) |
| 31 | \( 1 + 26.9T + 961T^{2} \) |
| 37 | \( 1 + 51.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 54.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 81.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 27.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 53.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 42.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 84.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 77.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 99.9iT - 9.40e3T^{2} \) |
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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
−8.306691789403805246149306907206, −7.23761276501284306226320787987, −6.86954789292832185096309968940, −5.67067031035106571216601875172, −5.28620553137757889557341355084, −4.47129803915907060807307275613, −3.52554858851317501366702238314, −2.38734590423591626240169954560, −1.83979752462376881280293752189, −0.41912133979473173241456692470,
0.956410754802383785093794389017, 1.65665246338275273482295543362, 3.29654199526339766317298133304, 3.65508158724606382013307966891, 4.26807855760561466626229621094, 5.72579960833324309546818636783, 6.16847847402239261730463275275, 6.68371154589573530243172759718, 7.944482395456020070466021844891, 8.158250714600162034500028686649
