| L(s) = 1 | + (−0.642 + 1.26i)2-s + (0.569 − 3.59i)3-s + (−1.17 − 1.61i)4-s + (2.10 − 4.53i)5-s + (4.16 + 3.02i)6-s + (0.635 + 0.635i)7-s + (2.79 − 0.442i)8-s + (−4.05 − 1.31i)9-s + (4.36 + 5.56i)10-s + (4.24 + 13.0i)11-s + (−6.48 + 3.30i)12-s + (−4.14 − 8.13i)13-s + (−1.20 + 0.392i)14-s + (−15.1 − 10.1i)15-s + (−1.23 + 3.80i)16-s + (−0.920 − 5.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (0.189 − 1.19i)3-s + (−0.293 − 0.404i)4-s + (0.421 − 0.906i)5-s + (0.694 + 0.504i)6-s + (0.0908 + 0.0908i)7-s + (0.349 − 0.0553i)8-s + (−0.450 − 0.146i)9-s + (0.436 + 0.556i)10-s + (0.385 + 1.18i)11-s + (−0.540 + 0.275i)12-s + (−0.318 − 0.625i)13-s + (−0.0863 + 0.0280i)14-s + (−1.00 − 0.677i)15-s + (−0.0772 + 0.237i)16-s + (−0.0541 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.796 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.00694 - 0.339201i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00694 - 0.339201i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 1.26i)T \) |
| 5 | \( 1 + (-2.10 + 4.53i)T \) |
| good | 3 | \( 1 + (-0.569 + 3.59i)T + (-8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (-0.635 - 0.635i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.24 - 13.0i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (4.14 + 8.13i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (0.920 + 5.81i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (18.5 - 25.4i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-27.4 - 14.0i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-33.4 - 46.0i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (1.57 + 1.14i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (10.4 - 5.33i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-20.6 + 63.6i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (38.8 - 38.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (49.2 + 7.80i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (5.69 - 35.9i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (57.8 + 18.7i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-4.37 - 13.4i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (10.1 + 63.8i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-25.4 + 18.4i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (61.7 + 31.4i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-33.4 - 45.9i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-40.0 + 6.33i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (57.1 - 18.5i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 1.94i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−15.14852002690285712143613127918, −14.04213318310166559672199963376, −12.83647964236724486449301804720, −12.29952579136373041758856249762, −10.15568089740040635892747850094, −8.847080452092156795221273022638, −7.72230369622734012764146838272, −6.58570565986156262140369390675, −4.99243184137468072333534855717, −1.58800842353009429692309085554,
2.94257835163018886009128878601, 4.46690544175616041891758364281, 6.57850948317007009341564708252, 8.635827149100724828137666561107, 9.661426607149182521830397177793, 10.70785460140791952550485886276, 11.40095089521175919925606034926, 13.27916164857791475656871599010, 14.40460361387456752075496849230, 15.34976405075780604228783134232
