| L(s) = 1 | + 0.192i·2-s + 3.96·4-s + 7.62i·5-s + 2.45·7-s + 1.53i·8-s − 1.46·10-s − 20.4·13-s + 0.473i·14-s + 15.5·16-s + 14.2i·17-s + 25.2·19-s + 30.2i·20-s − 7.64i·23-s − 33.1·25-s − 3.94i·26-s + ⋯ |
| L(s) = 1 | + 0.0963i·2-s + 0.990·4-s + 1.52i·5-s + 0.350·7-s + 0.191i·8-s − 0.146·10-s − 1.57·13-s + 0.0338i·14-s + 0.972·16-s + 0.841i·17-s + 1.32·19-s + 1.51i·20-s − 0.332i·23-s − 1.32·25-s − 0.151i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.122698845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122698845\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.192iT - 4T^{2} \) |
| 5 | \( 1 - 7.62iT - 25T^{2} \) |
| 7 | \( 1 - 2.45T + 49T^{2} \) |
| 13 | \( 1 + 20.4T + 169T^{2} \) |
| 17 | \( 1 - 14.2iT - 289T^{2} \) |
| 19 | \( 1 - 25.2T + 361T^{2} \) |
| 23 | \( 1 + 7.64iT - 529T^{2} \) |
| 29 | \( 1 - 43.4iT - 841T^{2} \) |
| 31 | \( 1 + 33.4T + 961T^{2} \) |
| 37 | \( 1 - 24.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 21.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 17.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 76.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.42iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 68.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 30.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 131.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
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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
−10.16019818672891459540111230519, −9.339621265820353204443423932544, −7.83558896844873069409121777882, −7.44659571361276894730727540685, −6.73805764555548587041984971480, −5.93810221916183636715579534861, −4.89927439430156005233743437692, −3.37340254019189160780599448033, −2.74432945297414111655168425868, −1.70551406709084650467953495909,
0.59669814716007707397166022965, 1.76919119218187547215694575914, 2.81119449229483724614143772792, 4.22440481795028536987002320984, 5.18827817383131075064321828516, 5.69965338243299239791131286708, 7.20053652481921346217880720482, 7.56634790787746648120203810239, 8.542279660495869697165332347588, 9.589598512682736090477573652473
