| L(s) = 1 | − 4-s + 4·11-s + 16-s + 16·19-s − 12·29-s + 2·31-s − 12·41-s − 4·44-s − 4·49-s + 18·59-s + 20·61-s − 5·64-s + 6·71-s − 16·76-s + 28·79-s + 6·89-s − 24·101-s + 40·109-s + 12·116-s + 10·121-s − 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.20·11-s + 1/4·16-s + 3.67·19-s − 2.22·29-s + 0.359·31-s − 1.87·41-s − 0.603·44-s − 4/7·49-s + 2.34·59-s + 2.56·61-s − 5/8·64-s + 0.712·71-s − 1.83·76-s + 3.15·79-s + 0.635·89-s − 2.38·101-s + 3.83·109-s + 1.11·116-s + 0.909·121-s − 0.179·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.344346179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.344346179\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 846 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 2856 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6006 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 181 T^{2} + 15312 T^{4} + 181 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 337 T^{2} + 47136 T^{4} + 337 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.37984371139627295707575306558, −6.25181260067699665293515135259, −5.89798203197987168623801166971, −5.47767788494491368543671941344, −5.47607893002501516626964292527, −5.30020533636251128982039308231, −5.12508699285880077556313900041, −5.01922970242236635655378679871, −4.88370421180243553454206742481, −4.19647423015583704088951166817, −4.18865673318681646381632944088, −4.07795559495342456142323845144, −3.68980322691500263352187302522, −3.48063205350727365954715845214, −3.37056978163305294804557508092, −3.25807119652633564179855378745, −2.93875926493938798921514204350, −2.45803285774228642410428663877, −2.36643584102057913008700606528, −1.78618455404133089448265138517, −1.73977134242938623385040232545, −1.50127805580789738618469791278, −0.929506163449034278337601152739, −0.64594502248579839485625231527, −0.59545285862512696724257669441,
0.59545285862512696724257669441, 0.64594502248579839485625231527, 0.929506163449034278337601152739, 1.50127805580789738618469791278, 1.73977134242938623385040232545, 1.78618455404133089448265138517, 2.36643584102057913008700606528, 2.45803285774228642410428663877, 2.93875926493938798921514204350, 3.25807119652633564179855378745, 3.37056978163305294804557508092, 3.48063205350727365954715845214, 3.68980322691500263352187302522, 4.07795559495342456142323845144, 4.18865673318681646381632944088, 4.19647423015583704088951166817, 4.88370421180243553454206742481, 5.01922970242236635655378679871, 5.12508699285880077556313900041, 5.30020533636251128982039308231, 5.47607893002501516626964292527, 5.47767788494491368543671941344, 5.89798203197987168623801166971, 6.25181260067699665293515135259, 6.37984371139627295707575306558
