| L(s) = 1 | − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s + 6·11-s + 2·14-s − 16-s + 6·17-s − 20-s − 6·22-s + 8·23-s + 25-s + 2·28-s + 4·29-s − 5·32-s − 6·34-s − 2·35-s − 4·37-s + 3·40-s − 2·43-s − 6·44-s − 8·46-s + 8·47-s − 3·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s + 1.80·11-s + 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.223·20-s − 1.27·22-s + 1.66·23-s + 1/5·25-s + 0.377·28-s + 0.742·29-s − 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.657·37-s + 0.474·40-s − 0.304·43-s − 0.904·44-s − 1.17·46-s + 1.16·47-s − 3/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.869866593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869866593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.16399751103832, −15.47257990165858, −14.64243959211847, −14.26818697844889, −13.87854692735768, −13.11272662172290, −12.65752680449208, −12.07802927678784, −11.41333201617904, −10.71048389731151, −10.02432593217566, −9.682951225770028, −9.161805288550961, −8.706585790936350, −8.090658814035643, −7.129098887693482, −6.837314628943155, −6.093636523863323, −5.304373770366630, −4.733967184017791, −3.678871812133920, −3.472829506572583, −2.274984101139950, −1.162684181867679, −0.8389474356698935,
0.8389474356698935, 1.162684181867679, 2.274984101139950, 3.472829506572583, 3.678871812133920, 4.733967184017791, 5.304373770366630, 6.093636523863323, 6.837314628943155, 7.129098887693482, 8.090658814035643, 8.706585790936350, 9.161805288550961, 9.682951225770028, 10.02432593217566, 10.71048389731151, 11.41333201617904, 12.07802927678784, 12.65752680449208, 13.11272662172290, 13.87854692735768, 14.26818697844889, 14.64243959211847, 15.47257990165858, 16.16399751103832
