| L(s) = 1 | + (2.82 + 4.89i)2-s + (21.8 + 12.6i)3-s + (−15.9 + 27.7i)4-s + (−10.7 + 6.20i)5-s + 142. i·6-s + (195. + 281. i)7-s − 181.·8-s + (−45.0 − 78.0i)9-s + (−60.8 − 35.1i)10-s + (774. − 1.34e3i)11-s + (−700. + 404. i)12-s − 2.77e3i·13-s + (−826. + 1.75e3i)14-s − 313.·15-s + (−512. − 886. i)16-s + (−109. − 63.2i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.810 + 0.468i)3-s + (−0.249 + 0.433i)4-s + (−0.0859 + 0.0496i)5-s + 0.661i·6-s + (0.570 + 0.821i)7-s − 0.353·8-s + (−0.0618 − 0.107i)9-s + (−0.0608 − 0.0351i)10-s + (0.582 − 1.00i)11-s + (−0.405 + 0.234i)12-s − 1.26i·13-s + (−0.301 + 0.639i)14-s − 0.0929·15-s + (−0.125 − 0.216i)16-s + (−0.0223 − 0.0128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.61547 + 1.17483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61547 + 1.17483i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.82 - 4.89i)T \) |
| 7 | \( 1 + (-195. - 281. i)T \) |
| good | 3 | \( 1 + (-21.8 - 12.6i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (10.7 - 6.20i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-774. + 1.34e3i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 2.77e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (109. + 63.2i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (1.14e3 - 662. i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-7.11e3 - 1.23e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 7.47e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (4.92e4 + 2.84e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-4.50e4 - 7.79e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 - 3.57e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 7.94e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-1.26e5 + 7.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-8.50e4 + 1.47e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.87e5 + 1.08e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.72e5 - 9.95e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (7.22e4 - 1.25e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.07e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (1.61e5 + 9.33e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (4.19e4 + 7.26e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 1.62e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (4.39e5 - 2.53e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 5.09e5iT - 8.32e11T^{2} \) |
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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
−18.47431590036296309230023420630, −17.02088447009539820656945343958, −15.35195121595322366432053633798, −14.80582362280487793566737420338, −13.34244475134478049288349462703, −11.51774494596687124557390066286, −9.228862072161692551368295724112, −8.052674674244446380541883275511, −5.66345889003333732125204759490, −3.37708825232127412988317115454,
1.89994797709191889328601166265, 4.30491369981107068926182868784, 7.23144016802011448639306366245, 9.045634725679316131435233218499, 10.89310432505141274252023487127, 12.47546478371239568211130371370, 13.93541279955924777309842395983, 14.62546262979703543457694544582, 16.79052970234637468099694268223, 18.35552707983146026976242494742
