| L(s) = 1 | − 8·5-s − 12·7-s + 44·11-s + 26·13-s − 56·17-s − 60·19-s + 168·23-s − 114·25-s − 216·29-s + 116·31-s + 96·35-s − 108·37-s − 464·41-s + 432·43-s + 308·47-s − 490·49-s − 416·53-s − 352·55-s + 148·59-s − 852·61-s − 208·65-s + 380·67-s − 860·71-s − 404·73-s − 528·77-s + 728·79-s − 1.09e3·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s − 0.647·7-s + 1.20·11-s + 0.554·13-s − 0.798·17-s − 0.724·19-s + 1.52·23-s − 0.911·25-s − 1.38·29-s + 0.672·31-s + 0.463·35-s − 0.479·37-s − 1.76·41-s + 1.53·43-s + 0.955·47-s − 1.42·49-s − 1.07·53-s − 0.862·55-s + 0.326·59-s − 1.78·61-s − 0.396·65-s + 0.692·67-s − 1.43·71-s − 0.647·73-s − 0.781·77-s + 1.03·79-s − 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3504384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.171168598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171168598\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 178 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 12 T + 634 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 86 p T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 9202 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 3970 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 168 T + 18718 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 216 T + 47770 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 116 T + 24138 T^{2} - 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 108 T + 14110 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 464 T + 187354 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 432 T + 163078 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 308 T + 227050 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 416 T + 340666 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 148 T + 253522 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 852 T + 465070 T^{2} + 852 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 380 T + 598818 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 860 T + 845722 T^{2} + 860 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 404 T + 616086 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 728 T + 307566 T^{2} - 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1092 T + 946690 T^{2} + 1092 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1792 T + 2197882 T^{2} - 1792 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 684 T + 1434022 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.912648666238608395715275536248, −8.902921733716050199424111499459, −8.281073546339059322219168207294, −7.958628741693447049171440367269, −7.39157479292450407604174587854, −7.12739095616712879146210117513, −6.63320909834483675107415026896, −6.42627984549524220529400092021, −5.89232823700541636274441888206, −5.63862862452443894890860570511, −4.74149378628280901363420889326, −4.62262067871413703320465491538, −4.03833329328810967555860041410, −3.68626123522913615522156940459, −3.22418992306099529151945928712, −2.88293077683429334838876597402, −1.88231142312022988005247909335, −1.74138147313860836409829207790, −0.874831035983803976234536379550, −0.27445774622771076669377282560,
0.27445774622771076669377282560, 0.874831035983803976234536379550, 1.74138147313860836409829207790, 1.88231142312022988005247909335, 2.88293077683429334838876597402, 3.22418992306099529151945928712, 3.68626123522913615522156940459, 4.03833329328810967555860041410, 4.62262067871413703320465491538, 4.74149378628280901363420889326, 5.63862862452443894890860570511, 5.89232823700541636274441888206, 6.42627984549524220529400092021, 6.63320909834483675107415026896, 7.12739095616712879146210117513, 7.39157479292450407604174587854, 7.958628741693447049171440367269, 8.281073546339059322219168207294, 8.902921733716050199424111499459, 8.912648666238608395715275536248
