| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 8.59i·5-s + 2.64·7-s + 2.82i·8-s + 12.1·10-s − 14.3i·11-s + 2.41·13-s − 3.74i·14-s + 4.00·16-s − 4.27i·17-s + 20.7·19-s − 17.1i·20-s − 20.3·22-s − 37.6i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.71i·5-s + 0.377·7-s + 0.353i·8-s + 1.21·10-s − 1.30i·11-s + 0.186·13-s − 0.267i·14-s + 0.250·16-s − 0.251i·17-s + 1.09·19-s − 0.859i·20-s − 0.923·22-s − 1.63i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.938744650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938744650\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 8.59iT - 25T^{2} \) |
| 11 | \( 1 + 14.3iT - 121T^{2} \) |
| 13 | \( 1 - 2.41T + 169T^{2} \) |
| 17 | \( 1 + 4.27iT - 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 + 37.6iT - 529T^{2} \) |
| 29 | \( 1 - 43.3iT - 841T^{2} \) |
| 31 | \( 1 - 45.5T + 961T^{2} \) |
| 37 | \( 1 - 4.78T + 1.36e3T^{2} \) |
| 41 | \( 1 - 2.97iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 1.53iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 127. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 2.38T + 5.32e3T^{2} \) |
| 79 | \( 1 + 48.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 85.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.88T + 9.40e3T^{2} \) |
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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
−9.940293915327812976499612900838, −8.836923428181563293753807861301, −8.077457043680697969837086764683, −7.08329266768108298590446448436, −6.31094622235655833149583614692, −5.37261461111897925227770374389, −4.08676971934920006091940185428, −3.06044124948853274179178167513, −2.61311716974978871634809976321, −0.954107722669007946823981261021,
0.804840386399014017307676013704, 1.88670135526998621317358233329, 3.80887270341030623406854913303, 4.70981066680268815097076212234, 5.20979975663662752792500933548, 6.12123536661514236389068417151, 7.38308353803528161078119510165, 7.952977983042808648718394216825, 8.676886239759160030237371117029, 9.693138797708717113781895148792
