| L(s) = 1 | + 2-s + 2·4-s − 2·5-s + 5·8-s − 2·10-s − 10·11-s − 5·13-s + 5·16-s − 3·17-s + 19-s − 4·20-s − 10·22-s − 6·23-s − 7·25-s − 5·26-s − 29-s + 10·32-s − 3·34-s − 3·37-s + 38-s − 10·40-s + 5·41-s + 43-s − 20·44-s − 6·46-s − 7·50-s − 10·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 0.894·5-s + 1.76·8-s − 0.632·10-s − 3.01·11-s − 1.38·13-s + 5/4·16-s − 0.727·17-s + 0.229·19-s − 0.894·20-s − 2.13·22-s − 1.25·23-s − 7/5·25-s − 0.980·26-s − 0.185·29-s + 1.76·32-s − 0.514·34-s − 0.493·37-s + 0.162·38-s − 1.58·40-s + 0.780·41-s + 0.152·43-s − 3.01·44-s − 0.884·46-s − 0.989·50-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9022964901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9022964901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−10.10975908493741980216243300731, −9.566058338771663774021175690797, −9.111269242153439718006318770987, −8.086348578748196574546208856336, −7.984976831339952332742512854429, −7.79423267258019069834091983942, −7.64713445959511679428753362097, −6.98336899778822456885614904066, −6.70681787520667898924656578083, −5.94838930988178074035359883457, −5.53314221970534617385585153080, −5.20490723814767860951649364375, −4.62129109571402038655602621534, −4.47613313249703904211666298643, −3.86162443369783560225439791103, −3.19136169758370000246267150629, −2.67449525611020922780945133742, −2.15312717684805437439347255666, −1.91758619685677832436011503217, −0.31409392486082828486863177249,
0.31409392486082828486863177249, 1.91758619685677832436011503217, 2.15312717684805437439347255666, 2.67449525611020922780945133742, 3.19136169758370000246267150629, 3.86162443369783560225439791103, 4.47613313249703904211666298643, 4.62129109571402038655602621534, 5.20490723814767860951649364375, 5.53314221970534617385585153080, 5.94838930988178074035359883457, 6.70681787520667898924656578083, 6.98336899778822456885614904066, 7.64713445959511679428753362097, 7.79423267258019069834091983942, 7.984976831339952332742512854429, 8.086348578748196574546208856336, 9.111269242153439718006318770987, 9.566058338771663774021175690797, 10.10975908493741980216243300731
