| L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 4·11-s + 4·15-s + 8·19-s + 4·21-s + 2·23-s + 3·25-s − 2·27-s − 4·31-s + 8·33-s + 4·35-s + 4·37-s + 4·41-s + 14·43-s + 10·47-s − 8·49-s − 16·53-s + 8·55-s + 16·57-s + 8·59-s − 4·61-s + 18·67-s + 4·69-s − 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 1.20·11-s + 1.03·15-s + 1.83·19-s + 0.872·21-s + 0.417·23-s + 3/5·25-s − 0.384·27-s − 0.718·31-s + 1.39·33-s + 0.676·35-s + 0.657·37-s + 0.624·41-s + 2.13·43-s + 1.45·47-s − 8/7·49-s − 2.19·53-s + 1.07·55-s + 2.11·57-s + 1.04·59-s − 0.512·61-s + 2.19·67-s + 0.481·69-s − 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.520336513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.520336513\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.537453025253144702234093302591, −9.505787242908214611883146459689, −8.994570710974860012793157397802, −8.915111815554237911515335869810, −8.119786568776193441008436788120, −8.089787924034208844889310672697, −7.35522326405815039229326843306, −7.23850417806235837933056963429, −6.61567147189225814031160257017, −6.07071273414998884812050220940, −5.65196166682403639269878644034, −5.38047197900176002481020886461, −4.64750200305293578317800206518, −4.33615390727861181813506908219, −3.60740115171368342337010128845, −3.24330185652132838407459964870, −2.63435167007807001565409260723, −2.26922215524656658901636912680, −1.41539907066851654198387062095, −1.07419388245125773622629065448,
1.07419388245125773622629065448, 1.41539907066851654198387062095, 2.26922215524656658901636912680, 2.63435167007807001565409260723, 3.24330185652132838407459964870, 3.60740115171368342337010128845, 4.33615390727861181813506908219, 4.64750200305293578317800206518, 5.38047197900176002481020886461, 5.65196166682403639269878644034, 6.07071273414998884812050220940, 6.61567147189225814031160257017, 7.23850417806235837933056963429, 7.35522326405815039229326843306, 8.089787924034208844889310672697, 8.119786568776193441008436788120, 8.915111815554237911515335869810, 8.994570710974860012793157397802, 9.505787242908214611883146459689, 9.537453025253144702234093302591
