L(s) = 1 | − 0.311i·2-s + 3.90·4-s + 5.94i·5-s − 8.67·7-s − 2.46i·8-s + 1.85·10-s − 18.6·13-s + 2.69i·14-s + 14.8·16-s − 9.59i·17-s − 5.76·19-s + 23.2i·20-s − 41.5i·23-s − 10.3·25-s + 5.82i·26-s + ⋯ |
L(s) = 1 | − 0.155i·2-s + 0.975·4-s + 1.18i·5-s − 1.23·7-s − 0.307i·8-s + 0.185·10-s − 1.43·13-s + 0.192i·14-s + 0.927·16-s − 0.564i·17-s − 0.303·19-s + 1.16i·20-s − 1.80i·23-s − 0.414·25-s + 0.223i·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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6308006842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6308006842\) |
\(L(2)\) |
|
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 | \( 1 + 0.311iT - 4T^{2} \) |
| 5 | \( 1 - 5.94iT - 25T^{2} \) |
| 7 | \( 1 + 8.67T + 49T^{2} \) |
| 13 | \( 1 + 18.6T + 169T^{2} \) |
| 17 | \( 1 + 9.59iT - 289T^{2} \) |
| 19 | \( 1 + 5.76T + 361T^{2} \) |
| 23 | \( 1 + 41.5iT - 529T^{2} \) |
| 29 | \( 1 - 17.3iT - 841T^{2} \) |
| 31 | \( 1 + 13.6T + 961T^{2} \) |
| 37 | \( 1 + 7.21T + 1.36e3T^{2} \) |
| 41 | \( 1 + 53.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 18.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 35.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 117.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 91.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 115. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 52.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 2.04T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 9.97T + 9.40e3T^{2} \) |
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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
−9.802146808313507034301835795110, −8.569323794693829060576078983655, −7.27007392969517236554219555398, −6.90297654369651894335643367749, −6.36599652906724425044544271377, −5.18973110611165255667330085483, −3.69983887406827689671101826900, −2.81393598041342808944366913281, −2.28622003426101014839589859493, −0.17297057741204928159720311492,
1.42154885705058961842055415158, 2.63393802918110670093716388319, 3.69104224531873942048306829789, 4.94527551678403581774077054026, 5.76224296028138893657859622702, 6.60467947369676739343299056490, 7.45469922307611906457636955774, 8.188847274626039771175410584242, 9.354615588762327038353897316333, 9.748317431224453604179579345642