| L(s) = 1 | + 9-s + 12·11-s + 4·16-s + 2·19-s + 32·29-s − 2·31-s − 24·41-s + 13·49-s − 16·59-s + 28·61-s + 24·71-s − 2·79-s − 24·89-s + 12·99-s + 20·101-s − 30·109-s + 58·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 3.61·11-s + 16-s + 0.458·19-s + 5.94·29-s − 0.359·31-s − 3.74·41-s + 13/7·49-s − 2.08·59-s + 3.58·61-s + 2.84·71-s − 0.225·79-s − 2.54·89-s + 1.20·99-s + 1.99·101-s − 2.87·109-s + 5.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.894986155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.894986155\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 90 T^{2} + 5891 T^{4} + 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 145 T^{2} + 15696 T^{4} + 145 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.102608414592475861344752028736, −7.49603697203859800003880980761, −7.38525620926607830956039300932, −6.82489351120759028108738776922, −6.61615645401728381683934064004, −6.61266813666980583828202704548, −6.58735462151392219376968625295, −6.54135486601632489052566134698, −5.91821278944175291492919135726, −5.70350134481724424894284273379, −5.18944761079774619361655332962, −5.02329368734033427578974993846, −4.98882105457079440425963807600, −4.51683677201848150417456923096, −4.05806247224254642935848743224, −3.95049740998953673640855107249, −3.91233971269841431758581248533, −3.40106121564142083238630934460, −3.13302237763329420687032780019, −2.78879427067648028942137348310, −2.45005423925989771272156664703, −1.86455872379287335964664445562, −1.33411197160609318755312097842, −1.05841596916921040601553479351, −1.00549029580644746002583426987,
1.00549029580644746002583426987, 1.05841596916921040601553479351, 1.33411197160609318755312097842, 1.86455872379287335964664445562, 2.45005423925989771272156664703, 2.78879427067648028942137348310, 3.13302237763329420687032780019, 3.40106121564142083238630934460, 3.91233971269841431758581248533, 3.95049740998953673640855107249, 4.05806247224254642935848743224, 4.51683677201848150417456923096, 4.98882105457079440425963807600, 5.02329368734033427578974993846, 5.18944761079774619361655332962, 5.70350134481724424894284273379, 5.91821278944175291492919135726, 6.54135486601632489052566134698, 6.58735462151392219376968625295, 6.61266813666980583828202704548, 6.61615645401728381683934064004, 6.82489351120759028108738776922, 7.38525620926607830956039300932, 7.49603697203859800003880980761, 8.102608414592475861344752028736
