| L(s) = 1 | + 4·5-s − 4·13-s + 8·25-s + 20·29-s + 12·37-s + 28·49-s − 4·53-s − 36·61-s − 16·65-s − 81-s + 32·97-s − 4·101-s − 28·109-s − 56·113-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.10·13-s + 8/5·25-s + 3.71·29-s + 1.97·37-s + 4·49-s − 0.549·53-s − 4.60·61-s − 1.98·65-s − 1/9·81-s + 3.24·97-s − 0.398·101-s − 2.68·109-s − 5.26·113-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.656570698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.656570698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^3$ | \( 1 - 238 T^{4} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 334 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
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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
−6.68577070027353049807163049405, −6.37900362857851601805402629439, −6.26503602321624040258843256656, −6.21856120997447912287217539023, −5.97097068535347759344926908460, −5.74363435358932667137850631067, −5.23902030478605624352511473934, −5.23146206846283663508559903941, −5.17362908854974470418697826315, −4.79951343323264380034690202662, −4.57699670522314557536258714918, −4.24398372583044967417854131304, −4.19767304473056575563759858033, −3.95705130539517658931747177999, −3.55369074677446455361106025394, −3.01546397300489061274900925217, −2.83222693585586459976081495873, −2.65827406220449919276072450302, −2.59645277391130277001907248782, −2.44617635815518350241150238947, −1.78497378311761638110817364505, −1.50081818716014592046802783145, −1.41877331146271692308526936889, −0.73796622959667285833522973722, −0.57842085269958824913690086723,
0.57842085269958824913690086723, 0.73796622959667285833522973722, 1.41877331146271692308526936889, 1.50081818716014592046802783145, 1.78497378311761638110817364505, 2.44617635815518350241150238947, 2.59645277391130277001907248782, 2.65827406220449919276072450302, 2.83222693585586459976081495873, 3.01546397300489061274900925217, 3.55369074677446455361106025394, 3.95705130539517658931747177999, 4.19767304473056575563759858033, 4.24398372583044967417854131304, 4.57699670522314557536258714918, 4.79951343323264380034690202662, 5.17362908854974470418697826315, 5.23146206846283663508559903941, 5.23902030478605624352511473934, 5.74363435358932667137850631067, 5.97097068535347759344926908460, 6.21856120997447912287217539023, 6.26503602321624040258843256656, 6.37900362857851601805402629439, 6.68577070027353049807163049405
