L(s) = 1 | + (−1.39 + 3.36i)5-s + (1.05 − 1.05i)7-s + (−1.50 + 3.64i)11-s + (−2.23 + 0.927i)13-s − 7.49·17-s + (−0.818 − 1.97i)19-s + (5.80 − 5.80i)23-s + (−5.86 − 5.86i)25-s + (0.326 − 0.135i)29-s − 1.71i·31-s + (2.08 + 5.03i)35-s + (0.387 + 0.160i)37-s + (−1.50 − 1.50i)41-s + (−7.40 − 3.06i)43-s + 7.27i·47-s + ⋯ |
L(s) = 1 | + (−0.624 + 1.50i)5-s + (0.399 − 0.399i)7-s + (−0.455 + 1.09i)11-s + (−0.620 + 0.257i)13-s − 1.81·17-s + (−0.187 − 0.453i)19-s + (1.21 − 1.21i)23-s + (−1.17 − 1.17i)25-s + (0.0606 − 0.0251i)29-s − 0.307i·31-s + (0.352 + 0.851i)35-s + (0.0636 + 0.0263i)37-s + (−0.234 − 0.234i)41-s + (−1.12 − 0.467i)43-s + 1.06i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3778430923\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3778430923\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.39 - 3.36i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.05 + 1.05i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.50 - 3.64i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (2.23 - 0.927i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 7.49T + 17T^{2} \) |
| 19 | \( 1 + (0.818 + 1.97i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.80 + 5.80i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.326 + 0.135i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.71iT - 31T^{2} \) |
| 37 | \( 1 + (-0.387 - 0.160i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (1.50 + 1.50i)T + 41iT^{2} \) |
| 43 | \( 1 + (7.40 + 3.06i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 7.27iT - 47T^{2} \) |
| 53 | \( 1 + (-3.94 - 1.63i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (12.7 + 5.30i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (0.579 + 1.40i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-7.96 + 3.29i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (4.75 + 4.75i)T + 71iT^{2} \) |
| 73 | \( 1 + (7.99 - 7.99i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + (1.35 - 0.560i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.75 - 4.75i)T - 89iT^{2} \) |
| 97 | \( 1 + 1.28T + 97T^{2} \) |
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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
−10.49757836436380690085253867850, −9.517157521206224765046838267194, −8.520026831673306182906511857040, −7.48539258932743092177014889554, −7.00292072766388855402732372193, −6.43504451273530880810374972528, −4.77708198410856324101960993323, −4.30723305887718099379883403309, −2.92337127132541470413576557485, −2.18165820652745466001246251408,
0.15733103712833601722993394022, 1.60084239634475251926962014518, 3.06183369680371733340473241860, 4.29032090208021960974043779294, 5.04429695564716460623626733338, 5.67849517267352497400048105962, 6.97783437303562615965858956660, 7.959756741697969920608613319179, 8.690551481139663667182176946052, 8.951238354868253897389594457877