| L(s) = 1 | − 4·7-s + 4·23-s + 8·31-s + 8·41-s − 20·47-s − 12·49-s + 8·71-s − 16·73-s + 32·79-s − 8·89-s + 16·97-s + 28·103-s − 24·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.834·23-s + 1.43·31-s + 1.24·41-s − 2.91·47-s − 1.71·49-s + 0.949·71-s − 1.87·73-s + 3.60·79-s − 0.847·89-s + 1.62·97-s + 2.75·103-s − 2.25·113-s + 3.27·121-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 + 0.0798·157-s − 1.26·161-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5417620483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5417620483\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3446 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 14294 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 11574 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.75934663024297788387983194931, −5.36573337468227404226047938845, −5.08407876486799013933822311377, −5.00022631833351081218365192928, −4.72709387336428966809989468205, −4.60521417854935242793493183796, −4.55985690545728024525193751384, −4.39513782897119132437353519885, −4.07902403975209386735752698549, −3.62648180011778115176262307797, −3.47763395559296021226023896024, −3.44974793824580270631494944663, −3.39685060049567578820883270883, −3.01982347968418382074957187267, −2.92421014296126352813731410517, −2.73830142421368923380013122667, −2.34091157185845732767230009187, −2.26240482520580481837288011335, −1.74409181101807960840247744029, −1.70296722530321838429599859013, −1.67581202925535317643983755860, −0.844582828836337705682473806206, −0.825083937110046974722486177524, −0.72127993693250806088246944841, −0.098979963737801417799850773461,
0.098979963737801417799850773461, 0.72127993693250806088246944841, 0.825083937110046974722486177524, 0.844582828836337705682473806206, 1.67581202925535317643983755860, 1.70296722530321838429599859013, 1.74409181101807960840247744029, 2.26240482520580481837288011335, 2.34091157185845732767230009187, 2.73830142421368923380013122667, 2.92421014296126352813731410517, 3.01982347968418382074957187267, 3.39685060049567578820883270883, 3.44974793824580270631494944663, 3.47763395559296021226023896024, 3.62648180011778115176262307797, 4.07902403975209386735752698549, 4.39513782897119132437353519885, 4.55985690545728024525193751384, 4.60521417854935242793493183796, 4.72709387336428966809989468205, 5.00022631833351081218365192928, 5.08407876486799013933822311377, 5.36573337468227404226047938845, 5.75934663024297788387983194931
