L(s) = 1 | + (−0.433 − 1.90i)2-s + (−2.52 + 1.21i)4-s + (0.623 + 0.781i)7-s + (2.19 + 2.74i)8-s + (0.974 + 0.222i)9-s + (−0.678 + 1.40i)11-s + (1.21 − 1.52i)14-s + (2.52 − 3.16i)16-s − 1.94i·18-s + (2.97 + 0.678i)22-s + (0.189 + 1.68i)23-s + (0.974 − 0.222i)25-s + (−2.52 − 1.21i)28-s + (−1.68 − 1.05i)29-s + (−3.94 − 1.90i)32-s + ⋯ |
L(s) = 1 | + (−0.433 − 1.90i)2-s + (−2.52 + 1.21i)4-s + (0.623 + 0.781i)7-s + (2.19 + 2.74i)8-s + (0.974 + 0.222i)9-s + (−0.678 + 1.40i)11-s + (1.21 − 1.52i)14-s + (2.52 − 3.16i)16-s − 1.94i·18-s + (2.97 + 0.678i)22-s + (0.189 + 1.68i)23-s + (0.974 − 0.222i)25-s + (−2.52 − 1.21i)28-s + (−1.68 − 1.05i)29-s + (−3.94 − 1.90i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 791 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7162986499\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7162986499\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.623 - 0.781i)T \) |
| 113 | \( 1 + (0.900 - 0.433i)T \) |
good | 2 | \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \) |
| 3 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 5 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 11 | \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 19 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 23 | \( 1 + (-0.189 - 1.68i)T + (-0.974 + 0.222i)T^{2} \) |
| 29 | \( 1 + (1.68 + 1.05i)T + (0.433 + 0.900i)T^{2} \) |
| 31 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 37 | \( 1 + (0.351 + 1.00i)T + (-0.781 + 0.623i)T^{2} \) |
| 41 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 43 | \( 1 + (-0.119 + 0.189i)T + (-0.433 - 0.900i)T^{2} \) |
| 47 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 53 | \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 61 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + (-1.00 + 0.351i)T + (0.781 - 0.623i)T^{2} \) |
| 71 | \( 1 + (-1.33 + 1.33i)T - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.467 + 1.33i)T + (-0.781 - 0.623i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (0.222 + 0.974i)T^{2} \) |
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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
−10.42266377987154448775725372965, −9.553164912230101076147312413869, −9.192288816216981559426067959566, −7.926489688398056592330754426070, −7.43481383720023319535156842428, −5.32561004763828664496565354451, −4.66351574668139823973008693054, −3.66975026455840992415552781272, −2.29018704340616531511241849411, −1.69724643302410716621258612141,
1.00171147182685974879987293577, 3.71980327707765271699251803048, 4.74777289983312382205070453882, 5.44359679047190599574061759909, 6.60378464918598083310774430048, 7.10104770411726318090203153909, 8.065062227165433533190741431518, 8.542724215194936189636475298718, 9.496004047213947895953582282017, 10.46328094487734294432843375578