| L(s) = 1 | + (−1.28 + 0.597i)2-s + (−1.23 + 1.21i)3-s + (1.28 − 1.53i)4-s + (−0.451 − 3.42i)5-s + (0.851 − 2.29i)6-s + (0.762 + 2.84i)7-s + (−0.734 + 2.73i)8-s + (0.0330 − 2.99i)9-s + (2.62 + 4.12i)10-s + (−2.71 + 2.08i)11-s + (0.281 + 3.45i)12-s + (−3.74 + 4.88i)13-s + (−2.67 − 3.19i)14-s + (4.73 + 3.67i)15-s + (−0.690 − 3.93i)16-s + 3.38·17-s + ⋯ |
| L(s) = 1 | + (−0.906 + 0.422i)2-s + (−0.710 + 0.703i)3-s + (0.643 − 0.765i)4-s + (−0.201 − 1.53i)5-s + (0.347 − 0.937i)6-s + (0.288 + 1.07i)7-s + (−0.259 + 0.965i)8-s + (0.0110 − 0.999i)9-s + (0.830 + 1.30i)10-s + (−0.819 + 0.628i)11-s + (0.0811 + 0.996i)12-s + (−1.03 + 1.35i)13-s + (−0.715 − 0.852i)14-s + (1.22 + 0.948i)15-s + (−0.172 − 0.984i)16-s + 0.820·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0200567 + 0.242510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0200567 + 0.242510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 - 0.597i)T \) |
| 3 | \( 1 + (1.23 - 1.21i)T \) |
| good | 5 | \( 1 + (0.451 + 3.42i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.762 - 2.84i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (2.71 - 2.08i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (3.74 - 4.88i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 3.38T + 17T^{2} \) |
| 19 | \( 1 + (2.96 - 1.22i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (5.41 + 1.44i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (6.01 + 0.791i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (0.785 - 0.453i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.444 + 1.07i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (3.23 - 12.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.21 - 2.88i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (4.40 + 2.54i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.68 - 6.48i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.339 - 0.0447i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.868 + 6.59i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (0.440 - 0.573i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-5.81 - 5.81i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.39 + 3.39i)T - 73iT^{2} \) |
| 79 | \( 1 + (-3.50 + 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.31 + 0.173i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-1.68 + 1.68i)T - 89iT^{2} \) |
| 97 | \( 1 + (-3.18 + 5.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.07278046307263182443655675517, −11.37881765366201288059884972871, −9.966166868555612509417545677568, −9.467566873278110672289135505620, −8.613086519428329316977842152806, −7.68954640717938711021142677375, −6.17497134864974326530268695274, −5.21668330442597988088566710135, −4.56535984444142875737817646658, −1.92594931623230911961771374623,
0.24854200501490085973303644058, 2.33648974344489442917084610368, 3.54839975298980911443629306315, 5.59826747453285006498732519111, 6.84080920002434447471948521539, 7.60862303237814740447595256686, 7.968636050683101076195084868968, 10.04571503945865407899548901295, 10.58023604837411640231323584493, 11.00617568914899857395403891265
