| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−5.12 − 2.95i)5-s + 2.82·8-s + (7.24 − 4.18i)10-s + (−0.878 − 1.52i)11-s + 18.7i·13-s + (−2.00 + 3.46i)16-s + (−20.3 + 11.7i)17-s + (19.9 + 11.5i)19-s + 11.8i·20-s + 2.48·22-s + (9.36 − 16.2i)23-s + (4.98 + 8.63i)25-s + (−22.9 − 13.2i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.02 − 0.591i)5-s + 0.353·8-s + (0.724 − 0.418i)10-s + (−0.0798 − 0.138i)11-s + 1.44i·13-s + (−0.125 + 0.216i)16-s + (−1.19 + 0.690i)17-s + (1.05 + 0.606i)19-s + 0.591i·20-s + 0.112·22-s + (0.407 − 0.705i)23-s + (0.199 + 0.345i)25-s + (−0.883 − 0.510i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8720793354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8720793354\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (5.12 + 2.95i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (0.878 + 1.52i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 18.7iT - 169T^{2} \) |
| 17 | \( 1 + (20.3 - 11.7i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-19.9 - 11.5i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-9.36 + 16.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + (-7.45 + 4.30i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-35.4 + 61.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.5 - 19.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (18.5 + 32.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-84.4 + 48.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (14.4 + 8.36i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (30.4 + 52.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (49.1 - 28.3i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-34.9 + 60.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (36.4 + 21.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 51.7iT - 9.40e3T^{2} \) |
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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
−9.516464154614292960425783305830, −8.960900790321200751128195375684, −8.185671697952066512988837792485, −7.40513554124193091514167633739, −6.60235926893671487906649371757, −5.56155294307697807657610488287, −4.42780002685877886887993365970, −3.86199587236489202408630599934, −1.99930189064364378008338899998, −0.45318610713704933851836536674,
0.865023435433351241376247024395, 2.67801784451326160488088142624, 3.32205760134875881783454661005, 4.44818494181542732715806952671, 5.48066156648677621799270363923, 6.91884221009537502781108332411, 7.55588158628215279108689362830, 8.262066243059981495946008444157, 9.298161958946662734923259612678, 10.03662067584300745613756617310
