| L(s) = 1 | + (0.907 − 1.08i)2-s + (−0.165 + 2.52i)3-s + (−0.352 − 1.96i)4-s + (−2.63 − 2.30i)5-s + (2.58 + 2.47i)6-s + (2.24 − 1.39i)7-s + (−2.45 − 1.40i)8-s + (−3.37 − 0.444i)9-s + (−4.89 + 0.759i)10-s + (3.51 + 1.73i)11-s + (5.03 − 0.564i)12-s + (−1.13 − 5.72i)13-s + (0.528 − 3.70i)14-s + (6.26 − 6.26i)15-s + (−3.75 + 1.38i)16-s + (1.44 − 5.39i)17-s + ⋯ |
| L(s) = 1 | + (0.641 − 0.766i)2-s + (−0.0955 + 1.45i)3-s + (−0.176 − 0.984i)4-s + (−1.17 − 1.03i)5-s + (1.05 + 1.00i)6-s + (0.849 − 0.526i)7-s + (−0.868 − 0.496i)8-s + (−1.12 − 0.148i)9-s + (−1.54 + 0.240i)10-s + (1.05 + 0.522i)11-s + (1.45 − 0.163i)12-s + (−0.315 − 1.58i)13-s + (0.141 − 0.989i)14-s + (1.61 − 1.61i)15-s + (−0.937 + 0.347i)16-s + (0.350 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0110 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0110 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.08620 - 1.07427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08620 - 1.07427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.907 + 1.08i)T \) |
| 7 | \( 1 + (-2.24 + 1.39i)T \) |
| good | 3 | \( 1 + (0.165 - 2.52i)T + (-2.97 - 0.391i)T^{2} \) |
| 5 | \( 1 + (2.63 + 2.30i)T + (0.652 + 4.95i)T^{2} \) |
| 11 | \( 1 + (-3.51 - 1.73i)T + (6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.13 + 5.72i)T + (-12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-1.44 + 5.39i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.47 + 1.17i)T + (15.0 + 11.5i)T^{2} \) |
| 23 | \( 1 + (-5.58 - 0.735i)T + (22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.50 + 2.24i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (1.48 - 0.857i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.49 - 2.18i)T + (4.82 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.30 - 5.57i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (4.72 + 7.06i)T + (-16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (1.28 + 4.79i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (4.58 - 9.28i)T + (-32.2 - 42.0i)T^{2} \) |
| 59 | \( 1 + (-4.05 - 11.9i)T + (-46.8 + 35.9i)T^{2} \) |
| 61 | \( 1 + (0.694 + 1.40i)T + (-37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (6.42 + 0.421i)T + (66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (-1.88 - 0.778i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.30 - 2.53i)T + (18.8 + 70.5i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 5.72i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-1.48 - 7.47i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-1.15 - 1.50i)T + (-23.0 + 85.9i)T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−11.00016625456566415073329005833, −10.16090363292727458271526079453, −9.303291704955110850572281139223, −8.480411886497034977703476028212, −7.25446514249997789155207260438, −5.35949040147962768738105616259, −4.71180429052329319934019441272, −4.21571098410888991585937187572, −3.18783586111123761066952956983, −0.850715822478758204536604362798,
1.96149084890305356011616679958, 3.47839587195159166214542402175, 4.52611392857898986299145189072, 6.19834403704397533450763618453, 6.64035783574789615796253255841, 7.44956892944713577412955603579, 8.192388860934459805224675906220, 8.926687734930797204921383031284, 11.12539241749809167995729535589, 11.52395833600035236516211114411
