| L(s) = 1 | + (0.330 + 0.943i)2-s + (0.166 − 0.476i)3-s + (−0.781 + 0.623i)4-s + (−1.39 − 1.74i)5-s + 0.505·6-s + (2.20 + 2.20i)7-s + (−0.846 − 0.532i)8-s + (2.14 + 1.71i)9-s + (1.19 − 1.89i)10-s + (1.86 − 2.33i)11-s + (0.166 + 0.476i)12-s + (0.817 − 1.30i)13-s + (−1.35 + 2.80i)14-s + (−1.06 + 0.372i)15-s + (0.222 − 0.974i)16-s + (−0.294 − 0.468i)17-s + ⋯ |
| L(s) = 1 | + (0.233 + 0.667i)2-s + (0.0963 − 0.275i)3-s + (−0.390 + 0.311i)4-s + (−0.623 − 0.781i)5-s + 0.206·6-s + (0.833 + 0.833i)7-s + (−0.299 − 0.188i)8-s + (0.715 + 0.570i)9-s + (0.376 − 0.598i)10-s + (0.561 − 0.703i)11-s + (0.0481 + 0.137i)12-s + (0.226 − 0.360i)13-s + (−0.361 + 0.750i)14-s + (−0.275 + 0.0962i)15-s + (0.0556 − 0.243i)16-s + (−0.0714 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.56170 + 0.512989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56170 + 0.512989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.330 - 0.943i)T \) |
| 5 | \( 1 + (1.39 + 1.74i)T \) |
| 43 | \( 1 + (4.57 + 4.69i)T \) |
| good | 3 | \( 1 + (-0.166 + 0.476i)T + (-2.34 - 1.87i)T^{2} \) |
| 7 | \( 1 + (-2.20 - 2.20i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.86 + 2.33i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.817 + 1.30i)T + (-5.64 - 11.7i)T^{2} \) |
| 17 | \( 1 + (0.294 + 0.468i)T + (-7.37 + 15.3i)T^{2} \) |
| 19 | \( 1 + (-3.02 - 3.78i)T + (-4.22 + 18.5i)T^{2} \) |
| 23 | \( 1 + (-3.39 + 0.382i)T + (22.4 - 5.11i)T^{2} \) |
| 29 | \( 1 + (-3.68 - 1.77i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-2.60 - 1.25i)T + (19.3 + 24.2i)T^{2} \) |
| 37 | \( 1 + (-3.71 - 3.71i)T + 37iT^{2} \) |
| 41 | \( 1 + (9.63 + 4.64i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (10.7 + 1.20i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (0.144 - 0.0909i)T + (22.9 - 47.7i)T^{2} \) |
| 59 | \( 1 + (-0.941 - 0.214i)T + (53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (0.350 + 0.727i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (6.00 + 0.676i)T + (65.3 + 14.9i)T^{2} \) |
| 71 | \( 1 + (-5.94 + 4.73i)T + (15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (11.3 + 7.10i)T + (31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 + (4.65 + 1.62i)T + (64.8 + 51.7i)T^{2} \) |
| 89 | \( 1 + (3.41 - 1.64i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (-1.47 - 13.0i)T + (-94.5 + 21.5i)T^{2} \) |
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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
−11.64586746857734892597599952037, −10.32562622218585840196969581792, −9.010053737804046849159512062102, −8.335648860036772807050461361157, −7.76961467579087240527793082135, −6.60371245559608458608481774929, −5.34498811725258809786209973338, −4.73540424808286512718196992946, −3.39976532511317625801283669772, −1.42877911961166021517384203544,
1.33062594417881082901912780863, 3.07252904539651731939856619976, 4.17287631420327491048601576162, 4.71647577427393822515395176494, 6.59773251244273242843515045275, 7.24632225757703042136944812982, 8.381470702223356198977572903927, 9.624838207479432358726558660062, 10.20540263833935205877743290202, 11.33221509282001013334018893309
