L(s) = 1 | + (−9.68 + 9.68i)3-s + (−49.1 + 26.5i)5-s + (−48.6 − 48.6i)7-s + 55.4i·9-s − 463. i·11-s + (320. + 320. i)13-s + (219. − 733. i)15-s + (1.04e3 − 1.04e3i)17-s − 701.·19-s + 942.·21-s + (2.00e3 − 2.00e3i)23-s + (1.71e3 − 2.61e3i)25-s + (−2.89e3 − 2.89e3i)27-s + 3.56e3i·29-s − 9.04e3i·31-s + ⋯ |
L(s) = 1 | + (−0.621 + 0.621i)3-s + (−0.879 + 0.475i)5-s + (−0.375 − 0.375i)7-s + 0.228i·9-s − 1.15i·11-s + (0.526 + 0.526i)13-s + (0.251 − 0.841i)15-s + (0.877 − 0.877i)17-s − 0.445·19-s + 0.466·21-s + (0.789 − 0.789i)23-s + (0.548 − 0.836i)25-s + (−0.762 − 0.762i)27-s + 0.787i·29-s − 1.69i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.605351 - 0.373512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.605351 - 0.373512i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (49.1 - 26.5i)T \) |
good | 3 | \( 1 + (9.68 - 9.68i)T - 243iT^{2} \) |
| 7 | \( 1 + (48.6 + 48.6i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 463. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-320. - 320. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-1.04e3 + 1.04e3i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 + 701.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.00e3 + 2.00e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 3.56e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 9.04e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.64e3 + 1.64e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (3.94e3 - 3.94e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (-7.94e3 - 7.94e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.16e4 - 1.16e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.12e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (9.19e3 + 9.19e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.26e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (2.79e4 + 2.79e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 8.22e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.72e4 + 7.72e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.45e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-9.78e4 + 9.78e4i)T - 8.58e9iT^{2} \) |
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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
−13.27137209659981804980346608682, −11.74083115476170051462862030937, −11.07024247383663761639081868267, −10.19242980793070151516109387264, −8.666162144396038815892064940788, −7.34450354559812604215727815640, −6.00229402962894823282771388938, −4.49361256639331849852771131290, −3.21924722325026178164662223178, −0.37522031494584147842910580944,
1.25480048903888908127183800244, 3.58683508466135063380423907859, 5.24247902948163695848532739532, 6.59134912984299217019755657488, 7.72765299845493955922843130673, 8.981414295530337157920059228510, 10.41179048980873477983091037720, 11.80424468542345968822594771583, 12.39826804788701776014263429534, 13.14633491857512244159881992363