| L(s) = 1 | + (0.474 − 0.311i)2-s + (−0.664 + 1.54i)4-s + (−1.99 + 2.11i)5-s + (−3.10 − 0.363i)7-s + (0.362 + 2.05i)8-s + (−0.286 + 1.62i)10-s + (0.984 − 3.28i)11-s + (0.231 − 3.98i)13-s + (−1.58 + 0.797i)14-s + (−1.48 − 1.57i)16-s + (0.878 − 0.319i)17-s + (−4.55 − 1.65i)19-s + (−1.92 − 4.47i)20-s + (−0.558 − 1.86i)22-s + (6.11 − 0.714i)23-s + ⋯ |
| L(s) = 1 | + (0.335 − 0.220i)2-s + (−0.332 + 0.770i)4-s + (−0.891 + 0.944i)5-s + (−1.17 − 0.137i)7-s + (0.128 + 0.726i)8-s + (−0.0905 + 0.513i)10-s + (0.296 − 0.991i)11-s + (0.0643 − 1.10i)13-s + (−0.424 + 0.213i)14-s + (−0.372 − 0.394i)16-s + (0.213 − 0.0775i)17-s + (−1.04 − 0.380i)19-s + (−0.431 − 1.00i)20-s + (−0.119 − 0.397i)22-s + (1.27 − 0.148i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0874418 - 0.192026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0874418 - 0.192026i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.474 + 0.311i)T + (0.792 - 1.83i)T^{2} \) |
| 5 | \( 1 + (1.99 - 2.11i)T + (-0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (3.10 + 0.363i)T + (6.81 + 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.984 + 3.28i)T + (-9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.231 + 3.98i)T + (-12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-0.878 + 0.319i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (4.55 + 1.65i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-6.11 + 0.714i)T + (22.3 - 5.30i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 0.807i)T + (17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-0.403 - 0.542i)T + (-8.89 + 29.6i)T^{2} \) |
| 37 | \( 1 + (8.73 + 7.32i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (5.55 + 3.65i)T + (16.2 + 37.6i)T^{2} \) |
| 43 | \( 1 + (2.17 - 0.515i)T + (38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (1.89 - 2.54i)T + (-13.4 - 45.0i)T^{2} \) |
| 53 | \( 1 + (4.18 - 7.25i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.44 - 4.81i)T + (-49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (0.226 + 0.523i)T + (-41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (12.2 - 6.13i)T + (40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (2.31 - 13.1i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.17 + 6.64i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.622 - 0.409i)T + (31.2 - 72.5i)T^{2} \) |
| 83 | \( 1 + (-3.04 + 1.99i)T + (32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (2.27 + 12.9i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (8.58 + 9.09i)T + (-5.64 + 96.8i)T^{2} \) |
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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
−10.40847165525802765925218560933, −9.028753379768215923418053240468, −8.408475637192331943584446458116, −7.36375356725257126551078080876, −6.74099602555606558903130473123, −5.58218603059430127156484431374, −4.20916890044917799054861672513, −3.17654254922737435641951125749, −3.04949651851535192072628988189, −0.098760020483074322550013926920,
1.58219583099180908653587415187, 3.53654244454341858181577502974, 4.46405509818535386428580434669, 5.07593663052466127004717744222, 6.49336646113904283902742858648, 6.83582800830491325535860416190, 8.268815827368832723187561118071, 9.113193262363087959200643986692, 9.687531449671091501612164069606, 10.55043753790879205823053946166
