| L(s) = 1 | + 4-s + 4·11-s + 4·16-s + 4·19-s − 8·29-s + 32·41-s + 4·44-s − 2·49-s − 12·59-s − 8·61-s + 11·64-s + 20·71-s + 4·76-s − 16·79-s + 4·89-s + 44·101-s − 28·109-s − 8·116-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.20·11-s + 16-s + 0.917·19-s − 1.48·29-s + 4.99·41-s + 0.603·44-s − 2/7·49-s − 1.56·59-s − 1.02·61-s + 11/8·64-s + 2.37·71-s + 0.458·76-s − 1.80·79-s + 0.423·89-s + 4.37·101-s − 2.68·109-s − 0.742·116-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.717783937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.717783937\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 302 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 24 T^{2} + 542 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 21 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5467 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 5003 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8014 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_4$ | \( ( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 146 T^{2} + 11427 T^{4} - 146 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 280 T^{2} + 30238 T^{4} - 280 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 15382 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 25798 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66775877760117130316455019265, −6.45480615872481844068539469859, −6.34974036175935273526693388067, −5.92866862813531010306566049673, −5.87415673104109862394886780134, −5.66944446697626137799834121124, −5.63997373239414771027463612164, −5.34859652009435925288220791751, −4.78898438152016987874951457191, −4.78525810740197747572049804826, −4.46162774579540810516331311285, −4.34571021059678637956435594192, −3.84825982488302644449214904540, −3.80661502230138959441136690908, −3.72067825028549213059396977759, −3.14145689669036212522323250713, −2.98185496620450859873366681031, −2.95901320591853251336578272933, −2.46882613788769181569860949530, −2.09217666317653287541762538032, −1.77616513749157110934777767466, −1.76052177084507313596685554946, −1.05143298270179940859915631036, −0.878081353727540340863022067729, −0.56978616255180607244449838896,
0.56978616255180607244449838896, 0.878081353727540340863022067729, 1.05143298270179940859915631036, 1.76052177084507313596685554946, 1.77616513749157110934777767466, 2.09217666317653287541762538032, 2.46882613788769181569860949530, 2.95901320591853251336578272933, 2.98185496620450859873366681031, 3.14145689669036212522323250713, 3.72067825028549213059396977759, 3.80661502230138959441136690908, 3.84825982488302644449214904540, 4.34571021059678637956435594192, 4.46162774579540810516331311285, 4.78525810740197747572049804826, 4.78898438152016987874951457191, 5.34859652009435925288220791751, 5.63997373239414771027463612164, 5.66944446697626137799834121124, 5.87415673104109862394886780134, 5.92866862813531010306566049673, 6.34974036175935273526693388067, 6.45480615872481844068539469859, 6.66775877760117130316455019265
