L(s) = 1 | + (0.190 + 0.587i)3-s + (−1.39 − 1.74i)5-s − 1.83·7-s + (2.11 − 1.53i)9-s + (4.19 + 3.04i)11-s + (2.22 − 1.61i)13-s + (0.757 − 1.15i)15-s + (2.02 − 6.23i)17-s + (2.20 − 6.79i)19-s + (−0.350 − 1.07i)21-s + (−3.57 − 2.59i)23-s + (−1.08 + 4.88i)25-s + (2.80 + 2.04i)27-s + (2.24 + 6.89i)29-s + (−0.240 + 0.740i)31-s + ⋯ |
L(s) = 1 | + (0.110 + 0.339i)3-s + (−0.626 − 0.779i)5-s − 0.692·7-s + (0.706 − 0.512i)9-s + (1.26 + 0.918i)11-s + (0.617 − 0.448i)13-s + (0.195 − 0.298i)15-s + (0.490 − 1.51i)17-s + (0.506 − 1.55i)19-s + (−0.0764 − 0.235i)21-s + (−0.745 − 0.541i)23-s + (−0.216 + 0.976i)25-s + (0.540 + 0.392i)27-s + (0.416 + 1.28i)29-s + (−0.0432 + 0.133i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25225 - 0.427281i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25225 - 0.427281i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.39 + 1.74i)T \) |
good | 3 | \( 1 + (-0.190 - 0.587i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 + (-4.19 - 3.04i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.22 + 1.61i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.02 + 6.23i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.20 + 6.79i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.57 + 2.59i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.24 - 6.89i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.240 - 0.740i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.60 + 1.89i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.84 - 2.06i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 + (1.96 + 6.03i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.74 - 8.44i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (0.345 - 0.250i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 1.13i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.0382 - 0.117i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.97 - 12.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.472 - 0.343i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.45 + 10.6i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (5.47 - 16.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.23 - 4.53i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.10 + 6.48i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.35595629250501435489309046278, −9.980918267792404340759480712883, −9.370776461308400299065933744719, −8.697523311805876774749776238303, −7.27088585477268545133103662185, −6.68573922035206344954939066692, −5.09454672525128985353358227877, −4.22237123527859644150776456892, −3.20333344415686612582544553280, −1.00982803517734375852602730982,
1.61256941855808855241610112693, 3.49393866008785515063155600057, 3.99185855558289963068375562003, 6.08939024618332743483920306777, 6.46890987167667580068999081868, 7.76593345729010661404165365737, 8.341657246932230555047168615503, 9.745098441552422654394950738976, 10.40201199099307545513475086865, 11.49664294178602316757665656589