| L(s) = 1 | + (−0.0871 − 0.996i)2-s + (−1.08 + 1.35i)3-s + (−0.984 + 0.173i)4-s + (1.26 + 1.84i)5-s + (1.44 + 0.962i)6-s + (2.58 + 0.691i)7-s + (0.258 + 0.965i)8-s + (−0.649 − 2.92i)9-s + (1.72 − 1.42i)10-s + (0.554 + 0.320i)11-s + (0.833 − 1.51i)12-s + (−1.44 − 3.10i)13-s + (0.463 − 2.63i)14-s + (−3.86 − 0.287i)15-s + (0.939 − 0.342i)16-s + (0.403 + 4.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.0616 − 0.704i)2-s + (−0.625 + 0.779i)3-s + (−0.492 + 0.0868i)4-s + (0.566 + 0.824i)5-s + (0.587 + 0.392i)6-s + (0.975 + 0.261i)7-s + (0.0915 + 0.341i)8-s + (−0.216 − 0.976i)9-s + (0.545 − 0.449i)10-s + (0.167 + 0.0965i)11-s + (0.240 − 0.438i)12-s + (−0.401 − 0.860i)13-s + (0.123 − 0.703i)14-s + (−0.997 − 0.0741i)15-s + (0.234 − 0.0855i)16-s + (0.0979 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07235 + 0.582137i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07235 + 0.582137i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0871 + 0.996i)T \) |
| 3 | \( 1 + (1.08 - 1.35i)T \) |
| 5 | \( 1 + (-1.26 - 1.84i)T \) |
| 19 | \( 1 + (0.973 - 4.24i)T \) |
| good | 7 | \( 1 + (-2.58 - 0.691i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.554 - 0.320i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 + 3.10i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.403 - 4.61i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-4.59 - 3.21i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.13 - 3.46i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.31 + 4.00i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.10 - 1.10i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.45 - 9.48i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-9.96 + 6.97i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (0.0396 + 0.00347i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (4.79 + 3.35i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (0.274 + 0.230i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 5.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.651 - 7.45i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (2.57 + 0.453i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.13 - 2.39i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-0.988 - 2.71i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (10.1 + 2.72i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (4.92 + 1.79i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-6.32 + 0.553i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.93554924352238876139659230991, −10.17818895848597486056578759655, −9.498823195261838111403595493336, −8.445117808863450283682122295651, −7.34779753702516685346493732400, −5.92018062610028420137249033225, −5.37151586599223316297316983760, −4.14484864673600451061065149635, −3.09993264542463391920005664857, −1.64865008639575824028292771781,
0.828928010765564098474474865494, 2.18215267447809898062503287558, 4.60157726294020696692265288353, 5.01272613930215177894387782257, 6.06747386343855488072769185790, 7.05590081697917973447701841928, 7.69764358154884285611121767585, 8.819360001299993100440478313037, 9.380211921880483641240991723696, 10.77845421336200326090799278509
