| L(s) = 1 | + 4.69i·3-s − 7.54·7-s − 13.0·9-s + (5.78 + 9.35i)11-s + 0.829·13-s − 29.8·17-s + 36.2i·19-s − 35.4i·21-s − 29.7i·23-s − 18.9i·27-s − 10.0i·29-s + 34.6·31-s + (−43.9 + 27.1i)33-s − 61.9i·37-s + 3.89i·39-s + ⋯ |
| L(s) = 1 | + 1.56i·3-s − 1.07·7-s − 1.44·9-s + (0.525 + 0.850i)11-s + 0.0637·13-s − 1.75·17-s + 1.90i·19-s − 1.68i·21-s − 1.29i·23-s − 0.700i·27-s − 0.346i·29-s + 1.11·31-s + (−1.33 + 0.822i)33-s − 1.67i·37-s + 0.0997i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1280470545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1280470545\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-5.78 - 9.35i)T \) |
| good | 3 | \( 1 - 4.69iT - 9T^{2} \) |
| 7 | \( 1 + 7.54T + 49T^{2} \) |
| 13 | \( 1 - 0.829T + 169T^{2} \) |
| 17 | \( 1 + 29.8T + 289T^{2} \) |
| 19 | \( 1 - 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 29.7iT - 529T^{2} \) |
| 29 | \( 1 + 10.0iT - 841T^{2} \) |
| 31 | \( 1 - 34.6T + 961T^{2} \) |
| 37 | \( 1 + 61.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 6.35iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 - 8.41iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 3.79T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 100.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 66.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + 1.65iT - 9.40e3T^{2} \) |
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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
−10.26679077724096028698783076477, −9.504409966148349380904380232306, −9.020687067797317515109649679606, −8.038284165090597561334951533051, −6.66260066126276707584620247124, −6.13448494901049594430829042898, −4.85783741765287495630120443299, −4.16719704454267118425100320862, −3.47325720447139442818769341629, −2.20441252033302037825847556763,
0.04127126036490997556572686159, 1.15171601573893355703250057925, 2.47423649687269157450535577678, 3.29360082146063257805740548206, 4.73987416139309872417334625371, 6.06558160944955002657608386101, 6.65978912557865608477282872148, 7.03340619435789790978400569699, 8.204741884641119503294506794401, 8.907987417359620775668691353318
