| L(s) = 1 | − 8·5-s − 14·7-s + 44·11-s − 60·13-s − 48·17-s − 80·19-s + 140·23-s − 158·25-s + 60·29-s − 232·31-s + 112·35-s − 180·37-s + 344·41-s + 224·43-s + 312·47-s + 147·49-s + 20·53-s − 352·55-s + 64·59-s + 204·61-s + 480·65-s − 872·67-s + 164·71-s + 1.50e3·73-s − 616·77-s − 1.64e3·79-s + 2.20e3·83-s + ⋯ |
| L(s) = 1 | − 0.715·5-s − 0.755·7-s + 1.20·11-s − 1.28·13-s − 0.684·17-s − 0.965·19-s + 1.26·23-s − 1.26·25-s + 0.384·29-s − 1.34·31-s + 0.540·35-s − 0.799·37-s + 1.31·41-s + 0.794·43-s + 0.968·47-s + 3/7·49-s + 0.0518·53-s − 0.862·55-s + 0.141·59-s + 0.428·61-s + 0.915·65-s − 1.59·67-s + 0.274·71-s + 2.40·73-s − 0.911·77-s − 2.33·79-s + 2.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.857152638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.857152638\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 8 T + 222 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 90 p T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 60 T + 2478 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48 T + 2966 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 80 T + 15142 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 140 T + 28838 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 60 T + 49502 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 232 T + 72862 T^{2} + 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 180 T + 64350 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 107190 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 224 T + 101158 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 312 T + 121982 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 20 T + 69758 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 64 T + 196182 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 204 T + 434622 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 872 T + 790038 T^{2} + 872 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 164 T + 140646 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1500 T + 1319238 T^{2} - 1500 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1640 T + 1417534 T^{2} + 1640 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2200 T + 2283174 T^{2} - 2200 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 264 T + 761686 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2092 T + 2771446 T^{2} - 2092 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.011875298247660104585361264006, −8.915155787253241000973850567404, −7.979256747634483238741519511550, −7.935140164245752350304911390129, −7.21808962905514237601983682641, −7.20966174171278189220056378462, −6.52665586459773461198053483033, −6.50485364300177150257173096665, −5.74189008875112337540754631050, −5.53119103897223756672093913133, −4.80102104101908921136216023494, −4.49864938271311853698205587488, −3.93245682685070615822222357716, −3.80403337645918182819828916313, −3.19880719570190457535587673294, −2.59816365032551859682360265172, −2.15209374453311559505112480709, −1.65495595929526907568772436625, −0.66291150523092815210643167378, −0.42347374496071095704724749735,
0.42347374496071095704724749735, 0.66291150523092815210643167378, 1.65495595929526907568772436625, 2.15209374453311559505112480709, 2.59816365032551859682360265172, 3.19880719570190457535587673294, 3.80403337645918182819828916313, 3.93245682685070615822222357716, 4.49864938271311853698205587488, 4.80102104101908921136216023494, 5.53119103897223756672093913133, 5.74189008875112337540754631050, 6.50485364300177150257173096665, 6.52665586459773461198053483033, 7.20966174171278189220056378462, 7.21808962905514237601983682641, 7.935140164245752350304911390129, 7.979256747634483238741519511550, 8.915155787253241000973850567404, 9.011875298247660104585361264006
