| L(s) = 1 | + (0.939 + 0.342i)3-s + (0.217 + 1.23i)5-s + (−2.78 − 1.60i)7-s + (0.766 + 0.642i)9-s + (5.43 − 3.14i)11-s + (0.166 + 0.457i)13-s + (−0.217 + 1.23i)15-s + (−1.40 + 1.17i)17-s + (4.29 + 0.763i)19-s + (−2.06 − 2.45i)21-s + (4.43 + 0.781i)23-s + (3.22 − 1.17i)25-s + (0.500 + 0.866i)27-s + (2.49 − 2.97i)29-s + (−1.51 + 2.63i)31-s + ⋯ |
| L(s) = 1 | + (0.542 + 0.197i)3-s + (0.0973 + 0.552i)5-s + (−1.05 − 0.606i)7-s + (0.255 + 0.214i)9-s + (1.63 − 0.946i)11-s + (0.0461 + 0.126i)13-s + (−0.0561 + 0.318i)15-s + (−0.339 + 0.285i)17-s + (0.984 + 0.175i)19-s + (−0.450 − 0.536i)21-s + (0.924 + 0.162i)23-s + (0.644 − 0.234i)25-s + (0.0962 + 0.166i)27-s + (0.463 − 0.552i)29-s + (−0.272 + 0.472i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.93939 + 0.167557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93939 + 0.167557i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (-4.29 - 0.763i)T \) |
| good | 5 | \( 1 + (-0.217 - 1.23i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.78 + 1.60i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.43 + 3.14i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.166 - 0.457i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.40 - 1.17i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-4.43 - 0.781i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.49 + 2.97i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.51 - 2.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.58iT - 37T^{2} \) |
| 41 | \( 1 + (2.94 - 8.09i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-10.8 + 1.90i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.18 + 2.60i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (6.95 + 1.22i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (2.39 - 2.01i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (1.09 - 6.21i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.42 + 7.06i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.17 + 12.3i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.820 + 0.298i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-12.1 - 4.43i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (13.4 + 7.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (2.12 + 5.82i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.95 - 7.10i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.01517431918640644690728520164, −9.273243797049502568647912181289, −8.673391223783251564915913501956, −7.46186855493147281354016498467, −6.63905846586792327890161662937, −6.11760213569131449637799010451, −4.55906547380020728863755121356, −3.49604478140343302529086098123, −3.01838501257382522691816972240, −1.17926299223077605496097583936,
1.19691084887354028961865104010, 2.57863914347404591773050374175, 3.62803878846885235697659271604, 4.66911891895265458527870172323, 5.80594323575261347143675054031, 6.81027777254622920033455897716, 7.34360970689736024592182461062, 8.761317306207913058072665593273, 9.260471818476449087989898277135, 9.585493784076406654724734827838
