| L(s) = 1 | + (1.56 + 0.732i)3-s + (0.407 + 0.341i)5-s + (0.507 + 0.184i)7-s + (1.92 + 2.29i)9-s + (−1.49 + 1.25i)11-s + (0.696 − 3.94i)13-s + (0.388 + 0.834i)15-s + (0.0114 − 0.0199i)17-s + (1.25 + 2.17i)19-s + (0.661 + 0.661i)21-s + (−6.43 + 2.34i)23-s + (−0.819 − 4.64i)25-s + (1.33 + 5.02i)27-s + (−1.03 − 5.88i)29-s + (3.81 − 1.38i)31-s + ⋯ |
| L(s) = 1 | + (0.906 + 0.423i)3-s + (0.182 + 0.152i)5-s + (0.191 + 0.0698i)7-s + (0.642 + 0.766i)9-s + (−0.449 + 0.377i)11-s + (0.193 − 1.09i)13-s + (0.100 + 0.215i)15-s + (0.00278 − 0.00483i)17-s + (0.287 + 0.498i)19-s + (0.144 + 0.144i)21-s + (−1.34 + 0.488i)23-s + (−0.163 − 0.929i)25-s + (0.257 + 0.966i)27-s + (−0.192 − 1.09i)29-s + (0.684 − 0.249i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57014 + 0.395269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57014 + 0.395269i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.56 - 0.732i)T \) |
| good | 5 | \( 1 + (-0.407 - 0.341i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.507 - 0.184i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (1.49 - 1.25i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.696 + 3.94i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0114 + 0.0199i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 - 2.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.43 - 2.34i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.03 + 5.88i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.81 + 1.38i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 6.21i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.30 - 7.42i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.23 - 3.55i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (10.4 + 3.81i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 + (3.02 + 2.53i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.15 + 3.33i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.88 + 10.6i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.30 + 3.98i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.36 - 14.4i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.23 - 12.6i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.334 + 1.89i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.68 - 6.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.32 - 1.95i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.57170122045692672616999808177, −11.32384078825475614662254953778, −10.08234954555724416908076450798, −9.752658243001566812278993556364, −8.148582497360882228891554065599, −7.86046423205781193148514448418, −6.16779208234457854696488374262, −4.85446050471168109743400273907, −3.54515036876401744311801455577, −2.21330950159023054067680571500,
1.75519026067211841880388355423, 3.26338523084797425322799996107, 4.65498883542547953783403398806, 6.23622750275082384582992172242, 7.29979219919612349095501479790, 8.339021221950373877848695992453, 9.137468953475989919898606752209, 10.12953377624078164946523980688, 11.39801215048237207368313457195, 12.34936079524081739209452054989
