| L(s) = 1 | + 2·7-s − 3·9-s + 4·16-s + 2·19-s − 8·31-s − 20·37-s − 11·49-s − 6·63-s + 32·67-s − 34·73-s + 28·97-s + 66·103-s − 4·109-s + 8·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s − 6·171-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 9-s + 16-s + 0.458·19-s − 1.43·31-s − 3.28·37-s − 1.57·49-s − 0.755·63-s + 3.90·67-s − 3.97·73-s + 2.84·97-s + 6.50·103-s − 0.383·109-s + 0.755·112-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.76·169-s − 0.458·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510824552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510824552\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 - 109 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + 17 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \) |
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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
−8.666686229138527669370415668387, −8.353078646265069113477089438741, −8.253363710990125608288182104209, −7.964564868613916924223372206597, −7.67632174342558231133387800131, −7.31777463497987380024673549703, −7.13188597114265530636648900114, −7.04849407572659673790749751181, −6.53858930006306273976638313649, −6.20667821347275736761351383920, −5.97467809956692366007062704781, −5.73926421859207841314585106419, −5.31382518620491011752720717662, −5.16593297280871508394435663282, −5.03176937330993522126408462253, −4.58114025526944434769903628398, −4.25880083933450224478574868539, −3.66550402670282311520218583431, −3.46092025716659151253838750154, −3.27393135133182694244412332112, −2.96980371316808273796371616269, −2.20513457695094955499927868300, −1.86332951529759131415440609928, −1.61023001958516016286348661855, −0.61155138404384136126541772739,
0.61155138404384136126541772739, 1.61023001958516016286348661855, 1.86332951529759131415440609928, 2.20513457695094955499927868300, 2.96980371316808273796371616269, 3.27393135133182694244412332112, 3.46092025716659151253838750154, 3.66550402670282311520218583431, 4.25880083933450224478574868539, 4.58114025526944434769903628398, 5.03176937330993522126408462253, 5.16593297280871508394435663282, 5.31382518620491011752720717662, 5.73926421859207841314585106419, 5.97467809956692366007062704781, 6.20667821347275736761351383920, 6.53858930006306273976638313649, 7.04849407572659673790749751181, 7.13188597114265530636648900114, 7.31777463497987380024673549703, 7.67632174342558231133387800131, 7.964564868613916924223372206597, 8.253363710990125608288182104209, 8.353078646265069113477089438741, 8.666686229138527669370415668387
