L(s) = 1 | + (−15.4 + 1.94i)3-s − 101.·5-s + 200. i·7-s + (235. − 60.0i)9-s + 350. i·11-s − 744. i·13-s + (1.56e3 − 196. i)15-s + 1.44e3i·17-s + 1.31e3·19-s + (−389. − 3.09e3i)21-s + 2.04e3·23-s + 7.09e3·25-s + (−3.52e3 + 1.38e3i)27-s + 3.78e3·29-s − 1.42e3i·31-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.124i)3-s − 1.80·5-s + 1.54i·7-s + (0.968 − 0.247i)9-s + 0.872i·11-s − 1.22i·13-s + (1.79 − 0.225i)15-s + 1.21i·17-s + 0.834·19-s + (−0.192 − 1.53i)21-s + 0.804·23-s + 2.27·25-s + (−0.930 + 0.366i)27-s + 0.835·29-s − 0.266i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8024797490\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8024797490\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (15.4 - 1.94i)T \) |
good | 5 | \( 1 + 101.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 200. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 350. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 744. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.44e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.42e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 2.61e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.08e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 3.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 539.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.37e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 5.04e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.57e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.32e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.58e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.73e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−11.08748910162305268207580334314, −10.18986921887159645126651957813, −8.943930837839687738652863829895, −8.000779098431943799156427774811, −7.21449544152026522697665462497, −5.96525310823983340024157257175, −5.06737704308579709459868689955, −4.09553105243087262428067340196, −2.85565833623841989502627638411, −0.969131108652346914174487411616,
0.39692012704761600418159544907, 0.932940830658210810541701664417, 3.38083957589660557209195093021, 4.24676700063401594430036732515, 4.99407473697915635398408756782, 6.72839763467754447605869776629, 7.19116769085818998294410140142, 7.962705440704539132253026853989, 9.283681949613612720719322575755, 10.57620813649173511902246447238