L(s) = 1 | + (−0.848 − 1.46i)2-s + (−0.978 − 0.207i)3-s + (−0.938 + 1.62i)4-s + (0.615 − 1.06i)5-s + (0.524 + 1.61i)6-s + 1.48·8-s + (0.913 + 0.406i)9-s − 2.08·10-s + (−0.961 − 1.66i)11-s + (1.25 − 1.39i)12-s + (−0.823 + 0.915i)15-s + (−0.322 − 0.558i)16-s + 1.53·17-s + (−0.177 − 1.68i)18-s + (1.15 + 2.00i)20-s + ⋯ |
L(s) = 1 | + (−0.848 − 1.46i)2-s + (−0.978 − 0.207i)3-s + (−0.938 + 1.62i)4-s + (0.615 − 1.06i)5-s + (0.524 + 1.61i)6-s + 1.48·8-s + (0.913 + 0.406i)9-s − 2.08·10-s + (−0.961 − 1.66i)11-s + (1.25 − 1.39i)12-s + (−0.823 + 0.915i)15-s + (−0.322 − 0.558i)16-s + 1.53·17-s + (−0.177 − 1.68i)18-s + (1.15 + 2.00i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4806894968\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4806894968\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.978 + 0.207i)T \) |
| 239 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.848 + 1.46i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.615 + 1.06i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.961 + 1.66i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.559 + 0.968i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.882 + 1.52i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.61T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.082241864589511149186815733247, −8.055920971070819355543742606790, −7.82027368839699837772481697192, −6.04903428444449485108085916564, −5.67200563971677939819412299362, −4.71764842793354407409246845164, −3.59877128804782167935158648578, −2.53565269838349848531665520969, −1.35439261655419051424549219296, −0.57387094503591662576650116101,
1.55625351009818551393931590265, 3.09097197010890830241674388559, 4.69819917730419610263238844993, 5.30808487161393889945833113243, 5.99705101908468550591145048353, 6.80259880996327575468592637154, 7.24865600570339856968895787588, 7.82531956156001582804836719658, 9.014368935415700441430079078607, 9.876769095871119192170116367731