L(s) = 1 | + (−0.595 + 1.03i)2-s + (1.52 − 2.63i)3-s + (0.290 + 0.503i)4-s + (1.81 + 3.14i)6-s + 0.609·7-s − 3.07·8-s + (−3.14 − 5.44i)9-s + 4.48·11-s + 1.77·12-s + (2.21 + 3.84i)13-s + (−0.362 + 0.628i)14-s + (1.24 − 2.16i)16-s + (1.45 − 2.51i)17-s + 7.48·18-s + (3.60 − 2.44i)19-s + ⋯ |
L(s) = 1 | + (−0.421 + 0.729i)2-s + (0.879 − 1.52i)3-s + (0.145 + 0.251i)4-s + (0.740 + 1.28i)6-s + 0.230·7-s − 1.08·8-s + (−1.04 − 1.81i)9-s + 1.35·11-s + 0.511·12-s + (0.615 + 1.06i)13-s + (−0.0969 + 0.167i)14-s + (0.312 − 0.540i)16-s + (0.352 − 0.609i)17-s + 1.76·18-s + (0.827 − 0.562i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67156 - 0.199441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67156 - 0.199441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.60 + 2.44i)T \) |
good | 2 | \( 1 + (0.595 - 1.03i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.52 + 2.63i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 0.609T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 + (-2.21 - 3.84i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.45 + 2.51i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.42 + 2.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.558 + 0.966i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 - 3.77T + 37T^{2} \) |
| 41 | \( 1 + (-4.15 + 7.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.99 - 8.65i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.94 + 5.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.22 - 7.31i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.11 - 8.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.49 - 4.31i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.23 - 7.34i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.80 - 10.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.86 + 3.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.51 - 7.82i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.83 - 8.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.50318777900986501428568226908, −9.381260468892437955985923750934, −8.956839352966792652718048058152, −8.135261508518811182522790357618, −7.22250633010283982380809716694, −6.80784026856440104731748521757, −5.92418117004596319141726006682, −3.87643831528172820034544591919, −2.67515084037100243000498400312, −1.32441814633975161070685139838,
1.63696031685538137958470236119, 3.24966801356794041294496311797, 3.73733934205580165168866284635, 5.18466572154839678082627803204, 6.15618941897532590999549231599, 7.896130240371761615714254614820, 8.722526392021306329443494033133, 9.559397712025654047492132786780, 9.956019079072939973617940596047, 10.90816974921579773340553938152