| L(s) = 1 | + 2-s + 6·5-s − 4·7-s − 8-s − 3·9-s + 6·10-s − 6·13-s − 4·14-s − 16-s + 12·17-s − 3·18-s + 19-s + 19·25-s − 6·26-s − 6·29-s + 12·34-s − 24·35-s + 38-s − 6·40-s − 3·41-s − 8·43-s − 18·45-s − 6·47-s − 2·49-s + 19·50-s + 6·53-s + 4·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.68·5-s − 1.51·7-s − 0.353·8-s − 9-s + 1.89·10-s − 1.66·13-s − 1.06·14-s − 1/4·16-s + 2.91·17-s − 0.707·18-s + 0.229·19-s + 19/5·25-s − 1.17·26-s − 1.11·29-s + 2.05·34-s − 4.05·35-s + 0.162·38-s − 0.948·40-s − 0.468·41-s − 1.21·43-s − 2.68·45-s − 0.875·47-s − 2/7·49-s + 2.68·50-s + 0.824·53-s + 0.534·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.673055844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673055844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 48 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 24 T + 271 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−13.92835200642126841447546786297, −13.26631228244715344175451384191, −12.98303247760488020421352443880, −12.55024267937959604068984170372, −12.02214196569113067546512546877, −11.44732127937116549735938712841, −10.16640429572740401937683306962, −10.15343836154507203378993261647, −9.649821626580960755746413617885, −9.491621912540238649998979429082, −8.708621184955583284032130744406, −7.75230985102298324916190910150, −6.97894429427254294578367153825, −6.28164355707810921589486350585, −5.76995324994470245826656273727, −5.47662583494971535382675742140, −4.98221653212859648678770452758, −3.24426703404144333301740499515, −3.05631423872479306912342985543, −1.95196628564662745654120187475,
1.95196628564662745654120187475, 3.05631423872479306912342985543, 3.24426703404144333301740499515, 4.98221653212859648678770452758, 5.47662583494971535382675742140, 5.76995324994470245826656273727, 6.28164355707810921589486350585, 6.97894429427254294578367153825, 7.75230985102298324916190910150, 8.708621184955583284032130744406, 9.491621912540238649998979429082, 9.649821626580960755746413617885, 10.15343836154507203378993261647, 10.16640429572740401937683306962, 11.44732127937116549735938712841, 12.02214196569113067546512546877, 12.55024267937959604068984170372, 12.98303247760488020421352443880, 13.26631228244715344175451384191, 13.92835200642126841447546786297
