| L(s) = 1 | + 4-s + 2·5-s + 9-s − 4·11-s + 12·19-s + 2·20-s + 5·25-s − 24·29-s − 4·31-s + 36-s − 8·41-s − 4·44-s + 2·45-s − 8·55-s + 16·59-s − 20·61-s − 64-s − 24·71-s + 12·76-s − 24·79-s + 20·89-s + 24·95-s − 4·99-s + 5·100-s + 20·101-s − 28·109-s − 24·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 2.75·19-s + 0.447·20-s + 25-s − 4.45·29-s − 0.718·31-s + 1/6·36-s − 1.24·41-s − 0.603·44-s + 0.298·45-s − 1.07·55-s + 2.08·59-s − 2.56·61-s − 1/8·64-s − 2.84·71-s + 1.37·76-s − 2.70·79-s + 2.11·89-s + 2.46·95-s − 0.402·99-s + 1/2·100-s + 1.99·101-s − 2.68·109-s − 2.22·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2050343942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2050343942\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 58 T^{2} + 1995 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.89619725752307381100975145102, −6.80300890410985513966541752420, −6.10811633797314390263028127198, −5.98663250134332642708621498983, −5.90483061202324443734226714843, −5.79420328283620025578242233687, −5.41704022637480138333897441283, −5.31383171730744051321805410474, −5.16306120676291996009026296036, −4.92771131404841427189294386469, −4.59554859236446153826338688743, −4.43864949452672669654151337814, −3.89021146762251261892014424131, −3.85002376590902417946788670153, −3.42953368669695346740675169272, −3.35546523042831410493943271320, −3.07249282525364232917865928460, −2.78330481029868093181720790861, −2.52826816062252476925996162429, −2.11489991981014125195562391935, −1.92193422188403187054799705015, −1.56408694582315674107207489478, −1.39794352721693562270798776356, −0.984692591527095093755742029180, −0.07870159420980363761312125416,
0.07870159420980363761312125416, 0.984692591527095093755742029180, 1.39794352721693562270798776356, 1.56408694582315674107207489478, 1.92193422188403187054799705015, 2.11489991981014125195562391935, 2.52826816062252476925996162429, 2.78330481029868093181720790861, 3.07249282525364232917865928460, 3.35546523042831410493943271320, 3.42953368669695346740675169272, 3.85002376590902417946788670153, 3.89021146762251261892014424131, 4.43864949452672669654151337814, 4.59554859236446153826338688743, 4.92771131404841427189294386469, 5.16306120676291996009026296036, 5.31383171730744051321805410474, 5.41704022637480138333897441283, 5.79420328283620025578242233687, 5.90483061202324443734226714843, 5.98663250134332642708621498983, 6.10811633797314390263028127198, 6.80300890410985513966541752420, 6.89619725752307381100975145102
