| L(s) = 1 | + (2.86 − 1.53i)3-s + (−0.865 − 1.05i)5-s + (−0.162 + 0.108i)7-s + (4.18 − 6.26i)9-s + (0.144 − 0.0437i)11-s + (−3.75 − 3.07i)13-s + (−4.09 − 1.69i)15-s + (−0.223 + 0.0925i)17-s + (0.742 + 7.53i)19-s + (−0.298 + 0.558i)21-s + (6.74 − 1.34i)23-s + (0.612 − 3.07i)25-s + (1.44 − 14.6i)27-s + (−1.92 + 6.33i)29-s + (2.37 + 2.37i)31-s + ⋯ |
| L(s) = 1 | + (1.65 − 0.883i)3-s + (−0.387 − 0.471i)5-s + (−0.0613 + 0.0409i)7-s + (1.39 − 2.08i)9-s + (0.0434 − 0.0131i)11-s + (−1.04 − 0.853i)13-s + (−1.05 − 0.437i)15-s + (−0.0541 + 0.0224i)17-s + (0.170 + 1.72i)19-s + (−0.0651 + 0.121i)21-s + (1.40 − 0.279i)23-s + (0.122 − 0.615i)25-s + (0.277 − 2.82i)27-s + (−0.356 + 1.17i)29-s + (0.427 + 0.427i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.77379 - 1.36463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77379 - 1.36463i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-2.86 + 1.53i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (0.865 + 1.05i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (0.162 - 0.108i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-0.144 + 0.0437i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (3.75 + 3.07i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (0.223 - 0.0925i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.742 - 7.53i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-6.74 + 1.34i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.92 - 6.33i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-2.37 - 2.37i)T + 31iT^{2} \) |
| 37 | \( 1 + (-3.51 - 0.346i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.456 - 2.29i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (4.58 + 2.45i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-3.90 - 9.42i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.703 + 2.32i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-1.33 + 1.09i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (3.57 + 6.69i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.30 + 2.43i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-3.30 - 4.93i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.33 + 5.57i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.333 + 0.804i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.71 + 0.169i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-8.77 - 1.74i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 - 5.19i)T + 97iT^{2} \) |
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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
−10.48772831337667278808141198458, −9.542614184300411986917437867148, −8.764437193177664996576398668155, −7.956533244457376931774648909187, −7.48232285071247728583413657721, −6.38282118904970588740752030480, −4.85051804489541212678021394560, −3.53330214112885095171763715501, −2.67909315420975505285552093872, −1.26953062167046046968290768808,
2.32110823440620125319359939095, 3.12037782751163645416474706401, 4.21632444907119937822952365543, 5.00343665126990745123127163001, 7.00155623401819182080667601456, 7.47055237006863628293894239300, 8.632008092301097677923230099111, 9.306684457049749895285314330429, 9.889700109779966520899495071397, 10.95306271550520724719091849876
