| L(s) = 1 | + (0.678 + 0.0974i)3-s + (−1.51 − 1.64i)5-s + (0.330 − 0.286i)7-s + (−2.42 − 0.712i)9-s + (−4.82 − 3.10i)11-s + (−1.83 − 1.59i)13-s + (−0.863 − 1.26i)15-s + (4.13 − 1.89i)17-s + (0.683 − 1.49i)19-s + (0.252 − 0.162i)21-s + (1.39 + 4.58i)23-s + (−0.435 + 4.98i)25-s + (−3.44 − 1.57i)27-s + (−2.82 − 6.18i)29-s + (0.516 + 3.58i)31-s + ⋯ |
| L(s) = 1 | + (0.391 + 0.0562i)3-s + (−0.675 − 0.737i)5-s + (0.124 − 0.108i)7-s + (−0.809 − 0.237i)9-s + (−1.45 − 0.934i)11-s + (−0.510 − 0.442i)13-s + (−0.222 − 0.326i)15-s + (1.00 − 0.458i)17-s + (0.156 − 0.343i)19-s + (0.0550 − 0.0353i)21-s + (0.290 + 0.956i)23-s + (−0.0871 + 0.996i)25-s + (−0.663 − 0.302i)27-s + (−0.524 − 1.14i)29-s + (0.0926 + 0.644i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.411400 - 0.756826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.411400 - 0.756826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.51 + 1.64i)T \) |
| 23 | \( 1 + (-1.39 - 4.58i)T \) |
| good | 3 | \( 1 + (-0.678 - 0.0974i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-0.330 + 0.286i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (4.82 + 3.10i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.83 + 1.59i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.13 + 1.89i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.683 + 1.49i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.82 + 6.18i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.516 - 3.58i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.55 + 8.68i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (5.57 - 1.63i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-10.5 - 1.51i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 12.5iT - 47T^{2} \) |
| 53 | \( 1 + (3.36 - 2.91i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.68 + 5.40i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.569 - 3.96i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-2.50 - 3.89i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-10.9 + 7.06i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (11.0 + 5.04i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.98 + 4.59i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (2.41 - 8.24i)T + (-69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (0.877 - 6.10i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.138 + 0.472i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.89046480509700574804728967500, −9.721630738710088443354072287126, −8.850307880306532145171741774080, −7.937398244841994617677921257210, −7.54240311842029332323839693619, −5.68545335783373922891871079722, −5.16083968115972127729480109752, −3.66064893515791814024466086751, −2.71948341385075124190227972353, −0.48394882464498225021523577046,
2.32414781331936453580902549482, 3.20544516170715998578803635975, 4.60234303087250620669346818163, 5.66069430046175088976083407364, 6.99915778543096537415063389341, 7.79854953563584946339789209963, 8.373699355619396969128070168642, 9.707999526764935202892310455103, 10.49422214548989229641786055241, 11.27955499124332930817379581636
