| L(s) = 1 | + 1.73i·2-s + 1.00·4-s + 4.89·5-s + 7.07i·7-s + 8.66i·8-s + 8.48i·10-s − 7.07i·13-s − 12.2·14-s − 10.9·16-s + 27.7i·17-s + 15.5i·19-s + 4.89·20-s − 24.4·23-s − 1.00·25-s + 12.2·26-s + ⋯ |
| L(s) = 1 | + 0.866i·2-s + 0.250·4-s + 0.979·5-s + 1.01i·7-s + 1.08i·8-s + 0.848i·10-s − 0.543i·13-s − 0.874·14-s − 0.687·16-s + 1.63i·17-s + 0.818i·19-s + 0.244·20-s − 1.06·23-s − 0.0400·25-s + 0.471·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.373171512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373171512\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73iT - 4T^{2} \) |
| 5 | \( 1 - 4.89T + 25T^{2} \) |
| 7 | \( 1 - 7.07iT - 49T^{2} \) |
| 13 | \( 1 + 7.07iT - 169T^{2} \) |
| 17 | \( 1 - 27.7iT - 289T^{2} \) |
| 19 | \( 1 - 15.5iT - 361T^{2} \) |
| 23 | \( 1 + 24.4T + 529T^{2} \) |
| 29 | \( 1 + 34.6iT - 841T^{2} \) |
| 31 | \( 1 - 12T + 961T^{2} \) |
| 37 | \( 1 + 60T + 1.36e3T^{2} \) |
| 41 | \( 1 - 34.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 83.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 83.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 48.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 26.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 120T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 128. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 97.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 70T + 9.40e3T^{2} \) |
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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
−10.20205737812999023365228953654, −8.905493496739081853588563071095, −8.361990609682695993796830847447, −7.55930149618672479362877101751, −6.39455647302500983901747122254, −5.77849199549530126489989344663, −5.52342807296158162970696981360, −3.95260112254731877192608546529, −2.46579614889713299279492429060, −1.79913029100093207890032394269,
0.66384147062550854152019800595, 1.81271981618786010618867717272, 2.71880750315588148020221549868, 3.80086102170529455164718539778, 4.81855192773414010500471084031, 5.94141426429298805107682272919, 7.01432571403536221231394501364, 7.29138878744174397356958899366, 8.884254225973689660705183461951, 9.542373254592768884632803915206
