| L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 10-s + 2·11-s + 4·13-s − 2·14-s − 16-s − 2·17-s − 20-s + 2·22-s + 4·23-s + 25-s + 4·26-s + 2·28-s + 4·29-s + 5·32-s − 2·34-s − 2·35-s − 3·40-s − 10·43-s − 2·44-s + 4·46-s − 12·47-s − 3·49-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 + 0.603·11-s + 1.10·13-s − 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s + 0.784·26-s + 0.377·28-s + 0.742·29-s + 0.883·32-s − 0.342·34-s − 0.338·35-s − 0.474·40-s − 1.52·43-s − 0.301·44-s + 0.589·46-s − 1.75·47-s − 3/7·49-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
| \(L(1)\) |
\(\approx\) |
\(2.444560313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444560313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.05924563120588, −15.18160833840029, −14.81326956056368, −14.17464142044492, −13.61785257454682, −13.20684858260365, −12.87018141750842, −12.20395209171754, −11.52781388379298, −11.03522288967006, −10.19073915353180, −9.644172310353046, −9.185795515474214, −8.545719862014391, −8.126671748494783, −6.852398507025408, −6.491339314076398, −6.084660061716592, −5.148974979978484, −4.791193369201695, −3.783983053590856, −3.458800072967541, −2.699829711093869, −1.602201411677286, −0.6188287410026561,
0.6188287410026561, 1.602201411677286, 2.699829711093869, 3.458800072967541, 3.783983053590856, 4.791193369201695, 5.148974979978484, 6.084660061716592, 6.491339314076398, 6.852398507025408, 8.126671748494783, 8.545719862014391, 9.185795515474214, 9.644172310353046, 10.19073915353180, 11.03522288967006, 11.52781388379298, 12.20395209171754, 12.87018141750842, 13.20684858260365, 13.61785257454682, 14.17464142044492, 14.81326956056368, 15.18160833840029, 16.05924563120588
