| L(s) = 1 | + 2·3-s + 2·5-s − 6·7-s + 3·9-s − 2·11-s − 6·13-s + 4·15-s + 2·19-s − 12·21-s − 12·23-s + 3·25-s + 4·27-s − 10·29-s − 4·31-s − 4·33-s − 12·35-s + 2·37-s − 12·39-s + 6·41-s − 6·43-s + 6·45-s − 20·47-s + 16·49-s − 4·55-s + 4·57-s − 12·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 2.26·7-s + 9-s − 0.603·11-s − 1.66·13-s + 1.03·15-s + 0.458·19-s − 2.61·21-s − 2.50·23-s + 3/5·25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s − 2.02·35-s + 0.328·37-s − 1.92·39-s + 0.937·41-s − 0.914·43-s + 0.894·45-s − 2.91·47-s + 16/7·49-s − 0.539·55-s + 0.529·57-s − 1.56·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 104 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 168 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−9.010016450796102564947735586508, −8.538688904736345477336043619962, −7.84327822722393718674513009035, −7.71948740898332032461388229645, −7.44128621014626699801732547193, −6.87969064854451314971609369608, −6.44341303902472256996622403150, −6.22477819396715579764507624959, −5.71505918988056270352534392382, −5.37551825403167024275416374702, −4.78862304978470796685144678068, −4.28502859918780940574269243520, −3.70681082271717832191169683704, −3.38718833782023405178767971351, −2.78645411329409699112232096660, −2.72802950633009112926375321664, −1.82377143662330237664447438244, −1.78812924739211803017917692003, 0, 0,
1.78812924739211803017917692003, 1.82377143662330237664447438244, 2.72802950633009112926375321664, 2.78645411329409699112232096660, 3.38718833782023405178767971351, 3.70681082271717832191169683704, 4.28502859918780940574269243520, 4.78862304978470796685144678068, 5.37551825403167024275416374702, 5.71505918988056270352534392382, 6.22477819396715579764507624959, 6.44341303902472256996622403150, 6.87969064854451314971609369608, 7.44128621014626699801732547193, 7.71948740898332032461388229645, 7.84327822722393718674513009035, 8.538688904736345477336043619962, 9.010016450796102564947735586508
