| L(s) = 1 | + (−0.669 + 0.743i)2-s + (0.0698 + 0.664i)3-s + (−0.104 − 0.994i)4-s + (−1.87 + 1.22i)5-s + (−0.540 − 0.392i)6-s + (2.57 − 0.622i)7-s + (0.809 + 0.587i)8-s + (2.49 − 0.531i)9-s + (0.345 − 2.20i)10-s + (2.69 + 0.572i)11-s + (0.653 − 0.138i)12-s + (−1.13 + 3.49i)13-s + (−1.25 + 2.32i)14-s + (−0.941 − 1.15i)15-s + (−0.978 + 0.207i)16-s + (−5.85 + 2.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (0.0403 + 0.383i)3-s + (−0.0522 − 0.497i)4-s + (−0.837 + 0.546i)5-s + (−0.220 − 0.160i)6-s + (0.971 − 0.235i)7-s + (0.286 + 0.207i)8-s + (0.832 − 0.177i)9-s + (0.109 − 0.698i)10-s + (0.811 + 0.172i)11-s + (0.188 − 0.0400i)12-s + (−0.314 + 0.968i)13-s + (−0.336 + 0.622i)14-s + (−0.243 − 0.299i)15-s + (−0.244 + 0.0519i)16-s + (−1.42 + 0.632i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.627483 + 0.799958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.627483 + 0.799958i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (1.87 - 1.22i)T \) |
| 7 | \( 1 + (-2.57 + 0.622i)T \) |
| good | 3 | \( 1 + (-0.0698 - 0.664i)T + (-2.93 + 0.623i)T^{2} \) |
| 11 | \( 1 + (-2.69 - 0.572i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (1.13 - 3.49i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (5.85 - 2.60i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (0.520 - 4.95i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (1.50 - 1.67i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (2.79 - 2.03i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.92 + 1.74i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-11.4 + 2.42i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-0.869 + 2.67i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (-7.77 - 3.46i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-1.43 - 13.6i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (7.66 + 8.51i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-7.38 + 8.19i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-5.87 + 2.61i)T + (44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (3.31 - 2.41i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (7.89 + 1.67i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (2.89 + 1.29i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-2.23 - 1.62i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-7.30 + 8.11i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.0317 + 0.0230i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.47603502046633489027122749611, −10.83880772614597807146870048722, −9.845869004234694753066926466038, −8.909396577957081695698025987925, −7.914553993318895529914147633814, −7.12207671492983522830473876177, −6.25400574132838452963458133618, −4.46575602956714166985290241899, −4.05654199029832582196449769695, −1.75792843055163506891169780002,
0.909402456899243916757876776467, 2.43810965972758826838110852366, 4.17423785411968633615524335213, 4.90946383963618945636532313232, 6.74078187623355220049203434187, 7.67096685135895216028094735145, 8.430256764500174696494207586869, 9.219467266929554197715689440627, 10.41401219756446318765923720067, 11.53538577428375810470661925606
