L(s) = 1 | + 2.34·2-s − 5.73·3-s + 1.51·4-s − 13.4·6-s − 9.56i·7-s − 5.83·8-s + 23.9·9-s + 4.15·11-s − 8.68·12-s − 4.55·13-s − 22.4i·14-s − 19.7·16-s + 27.9i·17-s + 56.1·18-s + (−7.04 + 17.6i)19-s + ⋯ |
L(s) = 1 | + 1.17·2-s − 1.91·3-s + 0.378·4-s − 2.24·6-s − 1.36i·7-s − 0.729·8-s + 2.65·9-s + 0.377·11-s − 0.723·12-s − 0.350·13-s − 1.60i·14-s − 1.23·16-s + 1.64i·17-s + 3.11·18-s + (−0.370 + 0.928i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0838 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0838 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7699449931\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7699449931\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (7.04 - 17.6i)T \) |
good | 2 | \( 1 - 2.34T + 4T^{2} \) |
| 3 | \( 1 + 5.73T + 9T^{2} \) |
| 7 | \( 1 + 9.56iT - 49T^{2} \) |
| 11 | \( 1 - 4.15T + 121T^{2} \) |
| 13 | \( 1 + 4.55T + 169T^{2} \) |
| 17 | \( 1 - 27.9iT - 289T^{2} \) |
| 23 | \( 1 + 13.9iT - 529T^{2} \) |
| 29 | \( 1 - 14.6iT - 841T^{2} \) |
| 31 | \( 1 - 28.9iT - 961T^{2} \) |
| 37 | \( 1 - 30.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 69.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 30.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 99.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.05T + 4.48e3T^{2} \) |
| 71 | \( 1 + 23.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 21.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 77.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 93.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 77.2T + 9.40e3T^{2} \) |
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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
−11.12647787281824765358511662659, −10.52918031988334388308718367332, −9.742746333649602529393067127697, −7.989380921841623714679515753335, −6.63512567469698832857336636442, −6.35016199860357657714387445818, −5.25429485362102600878589507959, −4.38441146407525277714004382232, −3.80064002533575368437758934808, −1.26469490719596338768522665166,
0.30219248620013287892202599633, 2.47406701086633123939018632866, 4.17336452513676768194999205233, 5.09763079744168741637301305512, 5.55898295999934421666583516674, 6.37442486960088555776130936442, 7.25120538837762670570562942843, 9.071716602254787131192046571817, 9.747070779883888504260658624568, 11.17496216125404633533607783188