| L(s) = 1 | + (−0.332 − 1.29i)2-s + (−2.49 + 0.475i)3-s + (0.186 − 0.102i)4-s + (−1.22 − 1.86i)5-s + (1.44 + 3.06i)6-s + (−2.17 + 2.99i)7-s + (−2.02 − 2.15i)8-s + (3.18 − 1.26i)9-s + (−2.01 + 2.20i)10-s + (−4.97 + 1.27i)11-s + (−0.416 + 0.344i)12-s + (0.0693 + 0.175i)13-s + (4.60 + 1.82i)14-s + (3.94 + 4.07i)15-s + (−1.89 + 2.97i)16-s + (2.79 − 5.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.235 − 0.915i)2-s + (−1.43 + 0.274i)3-s + (0.0933 − 0.0513i)4-s + (−0.548 − 0.836i)5-s + (0.589 + 1.25i)6-s + (−0.822 + 1.13i)7-s + (−0.715 − 0.762i)8-s + (1.06 − 0.421i)9-s + (−0.636 + 0.698i)10-s + (−1.49 + 0.385i)11-s + (−0.120 + 0.0993i)12-s + (0.0192 + 0.0486i)13-s + (1.23 + 0.487i)14-s + (1.01 + 1.05i)15-s + (−0.472 + 0.744i)16-s + (0.678 − 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0436890 + 0.211525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0436890 + 0.211525i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.22 + 1.86i)T \) |
| good | 2 | \( 1 + (0.332 + 1.29i)T + (-1.75 + 0.963i)T^{2} \) |
| 3 | \( 1 + (2.49 - 0.475i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (2.17 - 2.99i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (4.97 - 1.27i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.0693 - 0.175i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-2.79 + 5.08i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-1.31 + 6.87i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (-0.511 - 0.0321i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (1.82 + 0.230i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (0.917 + 0.504i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (6.38 + 4.05i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.356 - 5.67i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-3.07 + 0.999i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.555 + 0.591i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (-3.40 - 1.60i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (7.66 + 9.26i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.405 - 6.44i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-0.681 - 5.39i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (0.557 + 0.523i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (4.32 + 3.58i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-0.680 - 3.56i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (0.735 + 0.140i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-2.53 + 3.06i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-1.86 + 14.7i)T + (-93.9 - 24.1i)T^{2} \) |
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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
−12.41009782520807051308426525013, −11.75166043412717726455114873678, −10.99786122218688730259810443487, −9.886569804863067611468770605804, −9.053647209770931123223727444093, −7.19245464986402894528183213866, −5.72285960644163346519469054696, −4.94787816171092070644481559571, −2.82877127163893962481875393037, −0.27206719869658080109705306614,
3.48041361479287455375445596776, 5.57303921117047399994529856141, 6.34811254081103098546460282801, 7.32670205678049668091374922803, 8.009116303439153781105588804166, 10.41608940976903119325982591213, 10.61649208201082196097573034799, 11.91248952082750560238297855388, 12.76031236102118202155753855167, 14.09267466648218845097541028214
