L(s) = 1 | + 3.01i·2-s − 4.83i·3-s − 5.10·4-s + 14.5·6-s − 6.15·7-s − 3.33i·8-s − 14.3·9-s + 5.09·11-s + 24.6i·12-s + 4.20i·13-s − 18.5i·14-s − 10.3·16-s + 1.29·17-s − 43.2i·18-s + (18.3 + 4.84i)19-s + ⋯ |
L(s) = 1 | + 1.50i·2-s − 1.61i·3-s − 1.27·4-s + 2.43·6-s − 0.879·7-s − 0.417i·8-s − 1.59·9-s + 0.463·11-s + 2.05i·12-s + 0.323i·13-s − 1.32i·14-s − 0.647·16-s + 0.0764·17-s − 2.40i·18-s + (0.966 + 0.255i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6431129814\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6431129814\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-18.3 - 4.84i)T \) |
good | 2 | \( 1 - 3.01iT - 4T^{2} \) |
| 3 | \( 1 + 4.83iT - 9T^{2} \) |
| 7 | \( 1 + 6.15T + 49T^{2} \) |
| 11 | \( 1 - 5.09T + 121T^{2} \) |
| 13 | \( 1 - 4.20iT - 169T^{2} \) |
| 17 | \( 1 - 1.29T + 289T^{2} \) |
| 23 | \( 1 + 37.1T + 529T^{2} \) |
| 29 | \( 1 - 42.9iT - 841T^{2} \) |
| 31 | \( 1 - 52.1iT - 961T^{2} \) |
| 37 | \( 1 - 41.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.13iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 59.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 57.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 61.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 54.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 44.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 13.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 134.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 23.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 21.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 75.9iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.59840817130302657935414342043, −10.02537777469270829173444966965, −8.919929899877811233278499574477, −8.145255672966199249157933978485, −7.32280927238637464916432687471, −6.61549806143744090482720725546, −6.18181000157586560686999936019, −5.04748432333452287778558840080, −3.25934998945564362599021205612, −1.56248802651072059072692550163,
0.25914113419313076584262371059, 2.37233387635282767297902704370, 3.59710942355645029458121469619, 3.95115836185748702013554403550, 5.18971335091731241089023897626, 6.37682022642536575128771858342, 8.102921535010839920321486253091, 9.369911779139246158223247028367, 9.784897852434605984654436601216, 10.15196841918382889177453788424