| L(s) = 1 | − 5.30i·3-s − 0.206i·7-s − 19.1·9-s + 15.0i·11-s − 11.6·13-s + 18.1·17-s + 19.3i·19-s − 1.09·21-s + 27.2i·23-s + 53.6i·27-s − 44.4·29-s + 20.3i·31-s + 79.6·33-s + 18.1·37-s + 62.0i·39-s + ⋯ |
| L(s) = 1 | − 1.76i·3-s − 0.0295i·7-s − 2.12·9-s + 1.36i·11-s − 0.899·13-s + 1.07·17-s + 1.02i·19-s − 0.0521·21-s + 1.18i·23-s + 1.98i·27-s − 1.53·29-s + 0.657i·31-s + 2.41·33-s + 0.489·37-s + 1.59i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8495160090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8495160090\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.30iT - 9T^{2} \) |
| 7 | \( 1 + 0.206iT - 49T^{2} \) |
| 11 | \( 1 - 15.0iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 - 18.1T + 289T^{2} \) |
| 19 | \( 1 - 19.3iT - 361T^{2} \) |
| 23 | \( 1 - 27.2iT - 529T^{2} \) |
| 29 | \( 1 + 44.4T + 841T^{2} \) |
| 31 | \( 1 - 20.3iT - 961T^{2} \) |
| 37 | \( 1 - 18.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 32.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.06iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 5.37iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 79.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 83.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.51iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 15.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 10.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.1T + 9.40e3T^{2} \) |
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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
−10.03797439772891393557691478610, −9.346389061779468604801147616785, −8.009247094928790964246259837808, −7.55600871711364804729649121283, −6.96889489303890563426014459686, −5.92231434120680956082472172114, −5.07120599242926733142273387454, −3.46868957315388694714585776217, −2.14888236591819552324899314748, −1.39946803803984614867414536093,
0.28598036489030180847748494044, 2.70905900288798372574988483851, 3.53435357688124898477127901244, 4.52467214719913719560104950485, 5.34559874009358095191875622366, 6.10797151853311187905249350782, 7.55594648766376562086133899181, 8.584969810895850241842735302380, 9.237622854895694434901319752198, 9.949941893220896879927322790327
