| 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
| \(L(2)\) |
\(\approx\) |
\(2.650164918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.650164918\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 684 T + 1434022 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.145802262897093197403581091646, −8.644054785491324264791761698843, −8.164651389729053288420877211121, −7.934443645068471352531605143014, −7.68290405311072902474538336398, −7.10061831436141253845213338292, −6.43037197773822141927867403422, −6.32743981278490144138719285302, −5.78366275052439601870626874148, −5.74763535301356381520756427143, −5.00736294209181984914737642352, −4.62582597919480796252973200879, −4.07604087558583240797733099486, −3.69366743423158997652602608756, −2.96834246161620011156434193081, −2.77949647536009523826931986814, −2.04216743382387408501489904590, −1.79743085641532931659034312279, −0.837844445562178959551874520853, −0.41592717515004042802041438895,
0.41592717515004042802041438895, 0.837844445562178959551874520853, 1.79743085641532931659034312279, 2.04216743382387408501489904590, 2.77949647536009523826931986814, 2.96834246161620011156434193081, 3.69366743423158997652602608756, 4.07604087558583240797733099486, 4.62582597919480796252973200879, 5.00736294209181984914737642352, 5.74763535301356381520756427143, 5.78366275052439601870626874148, 6.32743981278490144138719285302, 6.43037197773822141927867403422, 7.10061831436141253845213338292, 7.68290405311072902474538336398, 7.934443645068471352531605143014, 8.164651389729053288420877211121, 8.644054785491324264791761698843, 9.145802262897093197403581091646
