| L(s) = 1 | + (−21.2 + 15.4i)3-s + 92.0·5-s + (26.8 + 82.7i)7-s + (139. − 428. i)9-s + (−196. − 605. i)11-s + (225. − 163. i)13-s + (−1.95e3 + 1.42e3i)15-s + (382. − 1.17e3i)17-s + (1.58e3 + 1.15e3i)19-s + (−1.85e3 − 1.34e3i)21-s + (726. − 2.23e3i)23-s + 5.33e3·25-s + (1.68e3 + 5.18e3i)27-s + (3.26e3 + 2.36e3i)29-s + (851. + 5.28e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.36 + 0.992i)3-s + 1.64·5-s + (0.207 + 0.638i)7-s + (0.572 − 1.76i)9-s + (−0.490 − 1.50i)11-s + (0.370 − 0.269i)13-s + (−2.24 + 1.63i)15-s + (0.320 − 0.987i)17-s + (1.00 + 0.732i)19-s + (−0.917 − 0.666i)21-s + (0.286 − 0.881i)23-s + 1.70·25-s + (0.444 + 1.36i)27-s + (0.720 + 0.523i)29-s + (0.159 + 0.987i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.652465608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.652465608\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (-851. - 5.28e3i)T \) |
| good | 3 | \( 1 + (21.2 - 15.4i)T + (75.0 - 231. i)T^{2} \) |
| 5 | \( 1 - 92.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-26.8 - 82.7i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 11 | \( 1 + (196. + 605. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-225. + 163. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-382. + 1.17e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.58e3 - 1.15e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-726. + 2.23e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.26e3 - 2.36e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 37 | \( 1 - 3.61e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (6.76e3 + 4.91e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + (-6.38e3 - 4.63e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (1.28e4 - 9.34e3i)T + (7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-5.26e3 + 1.62e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.92e4 - 1.39e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + 746.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.99e3 - 9.21e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-2.22e4 - 6.84e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.38e4 + 7.35e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (8.79e4 + 6.38e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-4.24e4 - 1.30e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (2.55e4 + 7.85e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.36296283208272606481656371372, −11.29397205858018039889869328633, −10.46566438359154012192826886306, −9.712280312671536195286237581661, −8.640936963000081626600798684143, −6.42956705762148367854547857334, −5.57445480701524352233087014025, −5.12882124625388400746226301377, −3.01541093217442268380881132996, −0.920109640101729953285081471992,
1.10127470176887020912928373462, 2.05058404761932187066212512791, 4.83328448240915202677483983782, 5.78766042734591785922744970024, 6.71232646976347282561942113930, 7.64043156498414692064720853583, 9.613263506330942215329265505863, 10.36927908379355844816064574662, 11.40742712580370264735046306314, 12.55420579474848866098686095730
