| L(s) = 1 | + 4·5-s + 9-s + 4·17-s + 6·25-s + 12·29-s − 8·37-s − 12·41-s + 4·45-s + 6·49-s − 4·53-s + 8·61-s + 12·73-s + 81-s + 16·85-s − 12·89-s − 12·97-s − 20·101-s + 16·109-s − 12·113-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1/3·9-s + 0.970·17-s + 6/5·25-s + 2.22·29-s − 1.31·37-s − 1.87·41-s + 0.596·45-s + 6/7·49-s − 0.549·53-s + 1.02·61-s + 1.40·73-s + 1/9·81-s + 1.73·85-s − 1.27·89-s − 1.21·97-s − 1.99·101-s + 1.53·109-s − 1.12·113-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.452785702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.452785702\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.375961952279118625968041857338, −8.932313246047740850729274066499, −8.224697628695080024457282408100, −8.130937759221178134675783859694, −7.11946169161695374029651407161, −6.75671098723169545747523830557, −6.40098553067410370307993344597, −5.68989039185040842224260471319, −5.34265058554374715577082422926, −4.90548385680552881991093212969, −4.09833843842773332368369623210, −3.30538674262792486177702740968, −2.65874373328413330863595718019, −1.89718091554560388544146377615, −1.20720397051797607053322546206,
1.20720397051797607053322546206, 1.89718091554560388544146377615, 2.65874373328413330863595718019, 3.30538674262792486177702740968, 4.09833843842773332368369623210, 4.90548385680552881991093212969, 5.34265058554374715577082422926, 5.68989039185040842224260471319, 6.40098553067410370307993344597, 6.75671098723169545747523830557, 7.11946169161695374029651407161, 8.130937759221178134675783859694, 8.224697628695080024457282408100, 8.932313246047740850729274066499, 9.375961952279118625968041857338
