L(s) = 1 | + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (−1.11 + 1.53i)17-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.190 − 0.587i)37-s + (0.5 − 1.53i)41-s + (−0.809 + 0.587i)45-s − 49-s + (−1.30 + 0.951i)53-s + (−1.11 + 0.363i)61-s + (−1.30 + 0.951i)65-s + (1.11 − 0.363i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)9-s + (0.5 − 1.53i)13-s + (−1.11 + 1.53i)17-s + (0.309 + 0.951i)25-s + (−1.11 − 1.53i)29-s + (0.190 − 0.587i)37-s + (0.5 − 1.53i)41-s + (−0.809 + 0.587i)45-s − 49-s + (−1.30 + 0.951i)53-s + (−1.11 + 0.363i)61-s + (−1.30 + 0.951i)65-s + (1.11 − 0.363i)73-s + (−0.809 − 0.587i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7805446435\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7805446435\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.809 + 0.587i)T \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−8.555591596230864262933126751342, −7.917620232005015055231186763684, −7.26678644220347364704891937833, −6.14458728011958769585507479192, −5.75377103737526064573708768654, −4.50680658835662007852365179417, −3.91920670958254897482281886614, −3.21046568866023685678916216702, −1.77613831638991402525053912138, −0.47057475450149827222893662640,
1.65884713252355986403527054651, 2.65872644772428954045324928006, 3.61135086686313717219857214790, 4.58167232460276288284455352005, 4.96704805545548452962532032260, 6.41693086560195444149693982333, 6.86168239651158523415909169088, 7.55192973477971436300001215294, 8.260576487814077029362469467109, 9.147200092802387740304888210730