L(s) = 1 | + (22.5 + 18.9i)3-s + (−48.8 + 17.7i)5-s + (−11.8 − 20.5i)7-s + (107. + 612. i)9-s + (−249. + 431. i)11-s + (58.3 − 48.9i)13-s + (−1.43e3 − 523. i)15-s + (282. − 1.59e3i)17-s + (−8.67 + 1.57e3i)19-s + (120. − 685. i)21-s + (215. + 78.2i)23-s + (−320. + 268. i)25-s + (−5.56e3 + 9.64e3i)27-s + (−663. − 3.76e3i)29-s + (3.29e3 + 5.71e3i)31-s + ⋯ |
L(s) = 1 | + (1.44 + 1.21i)3-s + (−0.874 + 0.318i)5-s + (−0.0913 − 0.158i)7-s + (0.444 + 2.51i)9-s + (−0.620 + 1.07i)11-s + (0.0957 − 0.0803i)13-s + (−1.64 − 0.600i)15-s + (0.236 − 1.34i)17-s + (−0.00551 + 0.999i)19-s + (0.0598 − 0.339i)21-s + (0.0847 + 0.0308i)23-s + (−0.102 + 0.0860i)25-s + (−1.46 + 2.54i)27-s + (−0.146 − 0.830i)29-s + (0.616 + 1.06i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.917848 + 1.91477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.917848 + 1.91477i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (8.67 - 1.57e3i)T \) |
good | 3 | \( 1 + (-22.5 - 18.9i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (48.8 - 17.7i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (11.8 + 20.5i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (249. - 431. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-58.3 + 48.9i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-282. + 1.59e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (-215. - 78.2i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (663. + 3.76e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-3.29e3 - 5.71e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.15e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.51e4 - 1.27e4i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (3.67e3 - 1.33e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (2.75e3 + 1.56e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (1.67e4 + 6.09e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (71.6 - 406. i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-4.09e4 - 1.48e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-5.36e3 - 3.04e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-5.04e4 + 1.83e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (1.99e4 + 1.67e4i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-2.96e4 - 2.48e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (6.30e3 + 1.09e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-6.29e4 + 5.27e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-1.56e4 + 8.85e4i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−14.25001785484912488989747310387, −13.09784251936863924924097790490, −11.53928269300586497330216152311, −10.21240352517359261158590008256, −9.561780180373268129886921150442, −8.152263401705652848790051142431, −7.43665720990392869614616248408, −4.86000865111437630660410621831, −3.77843168703841229953636068507, −2.58709161992189134149339910944,
0.792381745356484340994220619705, 2.58081772598901249548502013949, 3.85007125137493106016646233503, 6.23141251321461398837251041337, 7.69125194819306708124281660812, 8.263248438619972353495132841544, 9.231819548108704245480877471801, 11.13447589566434307788797549944, 12.46155183136264280566930966482, 13.10508287480884836348292483167