| L(s) = 1 | − 3-s + 5-s + 3·9-s + 11-s − 10·13-s − 15-s − 17-s + 6·19-s + 4·23-s − 8·27-s + 6·29-s − 2·31-s − 33-s − 8·37-s + 10·39-s − 20·41-s − 4·43-s + 3·45-s + 7·47-s + 51-s + 2·53-s + 55-s − 6·57-s − 14·59-s + 8·61-s − 10·65-s − 14·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 9-s + 0.301·11-s − 2.77·13-s − 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.834·23-s − 1.53·27-s + 1.11·29-s − 0.359·31-s − 0.174·33-s − 1.31·37-s + 1.60·39-s − 3.12·41-s − 0.609·43-s + 0.447·45-s + 1.02·47-s + 0.140·51-s + 0.274·53-s + 0.134·55-s − 0.794·57-s − 1.82·59-s + 1.02·61-s − 1.24·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.272899264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272899264\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18532959482445216862770727769, −9.765994721157904116722209607402, −9.637486934578038722909565588879, −9.028811615725359930153200547915, −8.725715283939267003927044676210, −7.938528198594060180773560421516, −7.60335019916407054816027280971, −7.07597207526028365304325457043, −6.95375780348520040040816799392, −6.58160469481264231171624965293, −5.79571053389078322178471225137, −5.19639133130327846629235103472, −5.15465273900548876370272694088, −4.65920022162677864775852065666, −4.14208950539746391445099325468, −3.20342815647747093083259013913, −3.02144290062743526287598137725, −1.95487639687081214359041701397, −1.73665222995988326943602610498, −0.53186778408631711985954219116,
0.53186778408631711985954219116, 1.73665222995988326943602610498, 1.95487639687081214359041701397, 3.02144290062743526287598137725, 3.20342815647747093083259013913, 4.14208950539746391445099325468, 4.65920022162677864775852065666, 5.15465273900548876370272694088, 5.19639133130327846629235103472, 5.79571053389078322178471225137, 6.58160469481264231171624965293, 6.95375780348520040040816799392, 7.07597207526028365304325457043, 7.60335019916407054816027280971, 7.938528198594060180773560421516, 8.725715283939267003927044676210, 9.028811615725359930153200547915, 9.637486934578038722909565588879, 9.765994721157904116722209607402, 10.18532959482445216862770727769
