| L(s) = 1 | + 2·3-s + 4·5-s + 2·9-s + 8·15-s − 12·19-s + 8·23-s + 6·27-s − 28·29-s − 20·43-s + 8·45-s − 32·47-s + 16·49-s + 4·53-s − 24·57-s − 12·67-s + 16·69-s + 8·71-s + 8·73-s + 11·81-s − 56·87-s − 48·95-s − 16·97-s + 20·101-s + 32·115-s + 16·121-s − 20·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 2/3·9-s + 2.06·15-s − 2.75·19-s + 1.66·23-s + 1.15·27-s − 5.19·29-s − 3.04·43-s + 1.19·45-s − 4.66·47-s + 16/7·49-s + 0.549·53-s − 3.17·57-s − 1.46·67-s + 1.92·69-s + 0.949·71-s + 0.936·73-s + 11/9·81-s − 6.00·87-s − 4.92·95-s − 1.62·97-s + 1.99·101-s + 2.98·115-s + 1.45·121-s − 1.78·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368055916\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 142 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 126 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 358 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4046 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2902 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_{4}$ | \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 13558 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7758 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 31774 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.34601987751179902385258766770, −7.27426726825701418649957522303, −6.89368360211312632214411479199, −6.78043772433461745121400803666, −6.40942945489643600095796549278, −6.33311189130423687249434381240, −5.94214237878264546495717936384, −5.94180968628171064633908720927, −5.53713211455925263401734154656, −5.28054154575686224755064032135, −5.05925155461602201975485784058, −4.85552592127439032413977388274, −4.58007398019946576177703920396, −4.24888606900431887658516334376, −3.76120586643914427108095633574, −3.61323731348774623382985820408, −3.50423865323254091183318992841, −3.26251381525882342521707204672, −2.63835358966363840630741313498, −2.34510526927726976772639631533, −2.27226121625915783222418550247, −1.74680060758197382820664442667, −1.61921106160247963726547859665, −1.55750505639739302578898936595, −0.22898443236149848169555542180,
0.22898443236149848169555542180, 1.55750505639739302578898936595, 1.61921106160247963726547859665, 1.74680060758197382820664442667, 2.27226121625915783222418550247, 2.34510526927726976772639631533, 2.63835358966363840630741313498, 3.26251381525882342521707204672, 3.50423865323254091183318992841, 3.61323731348774623382985820408, 3.76120586643914427108095633574, 4.24888606900431887658516334376, 4.58007398019946576177703920396, 4.85552592127439032413977388274, 5.05925155461602201975485784058, 5.28054154575686224755064032135, 5.53713211455925263401734154656, 5.94180968628171064633908720927, 5.94214237878264546495717936384, 6.33311189130423687249434381240, 6.40942945489643600095796549278, 6.78043772433461745121400803666, 6.89368360211312632214411479199, 7.27426726825701418649957522303, 7.34601987751179902385258766770
