L(s) = 1 | + (−0.642 − 0.766i)3-s + (−1.32 − 0.766i)7-s + (−0.173 + 0.984i)9-s + (1.43 + 1.20i)13-s + (0.866 + 0.5i)19-s + (0.266 + 1.50i)21-s + (−0.766 − 0.642i)25-s + (0.866 − 0.500i)27-s + (−0.984 + 1.70i)31-s − 0.347·37-s − 1.87i·39-s + (1.85 + 0.673i)43-s + (0.673 + 1.16i)49-s + (−0.173 − 0.984i)57-s + (−0.233 − 0.642i)61-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)3-s + (−1.32 − 0.766i)7-s + (−0.173 + 0.984i)9-s + (1.43 + 1.20i)13-s + (0.866 + 0.5i)19-s + (0.266 + 1.50i)21-s + (−0.766 − 0.642i)25-s + (0.866 − 0.500i)27-s + (−0.984 + 1.70i)31-s − 0.347·37-s − 1.87i·39-s + (1.85 + 0.673i)43-s + (0.673 + 1.16i)49-s + (−0.173 − 0.984i)57-s + (−0.233 − 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8589919230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8589919230\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.642 + 0.766i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.984 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 0.347T + T^{2} \) |
| 41 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-1.85 - 0.673i)T + (0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (0.233 + 0.642i)T + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-1.50 - 0.266i)T + (0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.984 + 0.826i)T + (0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \) |
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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
−8.699141105932810550138523716490, −7.71822678008044682023946297466, −7.11099514748031309264415018121, −6.35727144008757807097610055848, −6.09070180916153554347760439000, −5.01100102992459820118576173451, −3.92107343129491631294701471598, −3.33587666705626730338404007984, −1.97009339524807957554052129403, −0.950599115236907538673685555720,
0.72737901158980575407243583007, 2.52624906439606882180360318015, 3.51815327704964034915516401013, 3.84828991238020668377972825080, 5.27182399153020446076259198398, 5.75701669967590271030506310561, 6.19223637774671910354702769445, 7.14896433396158188901825102817, 8.098179275126308603785808806032, 9.064505269660328640949422991978