| L(s) = 1 | + (−1.30 − 0.814i)2-s + (1.85 − 0.531i)3-s + (0.158 + 0.324i)4-s + (−0.982 − 2.00i)5-s + (−2.84 − 0.816i)6-s + (0.367 − 0.637i)7-s + (−0.263 + 2.50i)8-s + (0.604 − 0.377i)9-s + (−0.355 + 3.41i)10-s + (1.34 − 1.49i)11-s + (0.466 + 0.517i)12-s + (−0.168 − 4.81i)13-s + (−0.998 + 0.530i)14-s + (−2.88 − 3.19i)15-s + (2.82 − 3.61i)16-s + (−1.55 + 0.108i)17-s + ⋯ |
| L(s) = 1 | + (−0.921 − 0.575i)2-s + (1.06 − 0.306i)3-s + (0.0792 + 0.162i)4-s + (−0.439 − 0.898i)5-s + (−1.16 − 0.333i)6-s + (0.139 − 0.240i)7-s + (−0.0930 + 0.885i)8-s + (0.201 − 0.125i)9-s + (−0.112 + 1.08i)10-s + (0.405 − 0.450i)11-s + (0.134 + 0.149i)12-s + (−0.0466 − 1.33i)13-s + (−0.266 + 0.141i)14-s + (−0.745 − 0.826i)15-s + (0.706 − 0.904i)16-s + (−0.377 + 0.0264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.127025 - 0.892086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.127025 - 0.892086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.982 + 2.00i)T \) |
| 19 | \( 1 + (4.25 + 0.953i)T \) |
| good | 2 | \( 1 + (1.30 + 0.814i)T + (0.876 + 1.79i)T^{2} \) |
| 3 | \( 1 + (-1.85 + 0.531i)T + (2.54 - 1.58i)T^{2} \) |
| 7 | \( 1 + (-0.367 + 0.637i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.34 + 1.49i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.168 + 4.81i)T + (-12.9 + 0.906i)T^{2} \) |
| 17 | \( 1 + (1.55 - 0.108i)T + (16.8 - 2.36i)T^{2} \) |
| 23 | \( 1 + (-0.267 + 0.0375i)T + (22.1 - 6.33i)T^{2} \) |
| 29 | \( 1 + (2.78 + 0.195i)T + (28.7 + 4.03i)T^{2} \) |
| 31 | \( 1 + (-8.77 + 3.90i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.0553 + 0.170i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.24 - 1.58i)T + (-9.91 - 39.7i)T^{2} \) |
| 43 | \( 1 + (1.18 + 6.73i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.58 + 0.530i)T + (46.5 + 6.54i)T^{2} \) |
| 53 | \( 1 + (-0.654 - 1.34i)T + (-32.6 + 41.7i)T^{2} \) |
| 59 | \( 1 + (-5.18 - 12.8i)T + (-42.4 + 40.9i)T^{2} \) |
| 61 | \( 1 + (-5.15 + 0.724i)T + (58.6 - 16.8i)T^{2} \) |
| 67 | \( 1 + (2.81 + 11.2i)T + (-59.1 + 31.4i)T^{2} \) |
| 71 | \( 1 + (-0.224 - 0.216i)T + (2.47 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-0.397 + 11.3i)T + (-72.8 - 5.09i)T^{2} \) |
| 79 | \( 1 + (-15.7 + 4.52i)T + (66.9 - 41.8i)T^{2} \) |
| 83 | \( 1 + (-5.48 + 2.44i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (5.66 + 7.25i)T + (-21.5 + 86.3i)T^{2} \) |
| 97 | \( 1 + (-0.373 + 1.49i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.49289949688435032589394756828, −9.504278432236862038391846577511, −8.670658171456935428659299779776, −8.291955943479557898174230030233, −7.55146075198431051918185347034, −5.87804748326163822726897384043, −4.65908065344573498724890741206, −3.27938451089217761210507960759, −2.05477416124915815111107184766, −0.65377608700167787543480608263,
2.21492871292670963857895867738, 3.54350179891410598671817521613, 4.35889107290971433540631345832, 6.47834769036102529348060689665, 6.92526279201619958707620648631, 8.111811012418572644700587659309, 8.556300885462035288378807751332, 9.468832774851378046022196286955, 10.05907384853096028394639073271, 11.28303373087428780150644911140
