| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + 2.23i·5-s − 1.06·7-s + 2.82i·8-s + 3.16·10-s + 0.743i·11-s − 3.34·13-s + 1.50i·14-s + 4.00·16-s − 27.5i·17-s + 6.98·19-s − 4.47i·20-s + 1.05·22-s + 31.9i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.447i·5-s − 0.151·7-s + 0.353i·8-s + 0.316·10-s + 0.0675i·11-s − 0.257·13-s + 0.107i·14-s + 0.250·16-s − 1.62i·17-s + 0.367·19-s − 0.223i·20-s + 0.0477·22-s + 1.38i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6981695643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6981695643\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 + 1.06T + 49T^{2} \) |
| 11 | \( 1 - 0.743iT - 121T^{2} \) |
| 13 | \( 1 + 3.34T + 169T^{2} \) |
| 17 | \( 1 + 27.5iT - 289T^{2} \) |
| 19 | \( 1 - 6.98T + 361T^{2} \) |
| 23 | \( 1 - 31.9iT - 529T^{2} \) |
| 29 | \( 1 - 48.9iT - 841T^{2} \) |
| 31 | \( 1 + 47.6T + 961T^{2} \) |
| 37 | \( 1 + 31.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 3.95iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.93T + 1.84e3T^{2} \) |
| 47 | \( 1 - 67.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.63iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 66.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 23.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 53.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 96.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 54.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 40.4T + 9.40e3T^{2} \) |
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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
−10.28097368530386303067545986126, −9.476608200000072264816730627427, −8.914579927345182233383591434395, −7.53748659789227394002086942787, −7.04851346880608601947509879349, −5.64233635929329736015265911662, −4.87309549588299135139500778715, −3.55290748280356609515732830131, −2.81063948627980814993746525638, −1.44812738972558057628546339892,
0.23653998284348764922500713510, 1.93014570941731228284661198040, 3.58313084179476109013421820327, 4.49181874821930126654352968674, 5.53925284201178411811381627454, 6.29808893952676854191668881825, 7.24017263704706710115552641107, 8.240570722787287986929595453417, 8.726414845104979603020106180327, 9.785710856710900509289637132436
