| L(s) = 1 | + (−31.9 − 2.05i)2-s − 192.·3-s + (1.01e3 + 131. i)4-s + (3.11e3 − 193. i)5-s + (6.13e3 + 393. i)6-s + (1.85e4 + 1.85e4i)7-s + (−3.21e4 − 6.26e3i)8-s − 2.21e4·9-s + (−9.99e4 − 226. i)10-s + (1.41e4 + 1.41e4i)11-s + (−1.95e5 − 2.51e4i)12-s − 5.58e5·13-s + (−5.54e5 − 6.30e5i)14-s + (−5.98e5 + 3.71e4i)15-s + (1.01e6 + 2.66e5i)16-s + (1.45e5 − 1.45e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0641i)2-s − 0.790·3-s + (0.991 + 0.127i)4-s + (0.998 − 0.0618i)5-s + (0.788 + 0.0506i)6-s + (1.10 + 1.10i)7-s + (−0.981 − 0.191i)8-s − 0.375·9-s + (−0.999 − 0.00226i)10-s + (0.0880 + 0.0880i)11-s + (−0.783 − 0.101i)12-s − 1.50·13-s + (−1.03 − 1.17i)14-s + (−0.788 + 0.0488i)15-s + (0.967 + 0.253i)16-s + (0.102 − 0.102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.00821437 - 0.0376328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00821437 - 0.0376328i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (31.9 + 2.05i)T \) |
| 5 | \( 1 + (-3.11e3 + 193. i)T \) |
| good | 3 | \( 1 + 192.T + 5.90e4T^{2} \) |
| 7 | \( 1 + (-1.85e4 - 1.85e4i)T + 2.82e8iT^{2} \) |
| 11 | \( 1 + (-1.41e4 - 1.41e4i)T + 2.59e10iT^{2} \) |
| 13 | \( 1 + 5.58e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + (-1.45e5 + 1.45e5i)T - 2.01e12iT^{2} \) |
| 19 | \( 1 + (-1.15e5 - 1.15e5i)T + 6.13e12iT^{2} \) |
| 23 | \( 1 + (7.04e6 - 7.04e6i)T - 4.14e13iT^{2} \) |
| 29 | \( 1 + (-4.93e6 - 4.93e6i)T + 4.20e14iT^{2} \) |
| 31 | \( 1 + 3.10e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 1.55e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.88e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.25e6iT - 2.16e16T^{2} \) |
| 47 | \( 1 + (-8.45e7 + 8.45e7i)T - 5.25e16iT^{2} \) |
| 53 | \( 1 + 3.51e5iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-1.96e8 + 1.96e8i)T - 5.11e17iT^{2} \) |
| 61 | \( 1 + (6.86e8 - 6.86e8i)T - 7.13e17iT^{2} \) |
| 67 | \( 1 + 7.27e7iT - 1.82e18T^{2} \) |
| 71 | \( 1 + 1.31e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + (1.78e9 - 1.78e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + 4.94e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 6.37e8T + 1.55e19T^{2} \) |
| 89 | \( 1 + 7.78e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + (1.05e10 - 1.05e10i)T - 7.37e19iT^{2} \) |
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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
−12.31695263903842215876201596129, −11.78602604239425405623893944572, −10.69443490177861167070801559956, −9.598332541750894733066127514071, −8.663244735090507940853774343477, −7.34807673026256176566276461022, −5.85348678041408417884739342035, −5.23748270366004190607038094702, −2.51143088942097861584019975277, −1.64337762413872723445863727875,
0.01580423443267285468052805476, 1.20880762275377288254722373713, 2.44488083636618721391472233663, 4.79864474888552049220289301857, 5.99379769514325349816024377298, 7.13779771086851537938691023225, 8.257647770039330076459898238628, 9.714496600532228931697079468121, 10.53075558043621472916649240650, 11.33261952041268380573687140485
