| L(s) = 1 | + 3-s − 2·5-s + 12·13-s − 2·15-s + 2·17-s − 4·19-s + 4·23-s + 5·25-s − 27-s − 20·29-s + 8·31-s − 6·37-s + 12·39-s − 4·41-s − 8·43-s − 8·47-s + 2·51-s + 10·53-s − 4·57-s − 12·59-s + 2·61-s − 24·65-s − 12·67-s + 4·69-s − 24·71-s + 14·73-s + 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 3.32·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 25-s − 0.192·27-s − 3.71·29-s + 1.43·31-s − 0.986·37-s + 1.92·39-s − 0.624·41-s − 1.21·43-s − 1.16·47-s + 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s + 0.256·61-s − 2.97·65-s − 1.46·67-s + 0.481·69-s − 2.84·71-s + 1.63·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.309745192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309745192\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−10.21562198516954790051202097404, −9.147582540727085318460012811390, −9.108091983481405883066704444257, −8.765441500504530385676685487280, −8.435841433342748560325439996034, −7.941271392306644231221917118094, −7.69948647082279765704915061134, −7.15881746019521794633840806448, −6.63142311653263516132064911194, −6.16947576792430642703492773079, −5.95264744156061957621713588411, −5.30343333664926211319715971269, −4.75194794536386495813638676141, −4.16533432858942224306190877309, −3.63816573635277172694295242339, −3.34638934955336140241032251799, −3.21493623792846808353999135188, −1.90499682676010143945605786879, −1.62505578711254294228561139434, −0.66220980053775235408078682433,
0.66220980053775235408078682433, 1.62505578711254294228561139434, 1.90499682676010143945605786879, 3.21493623792846808353999135188, 3.34638934955336140241032251799, 3.63816573635277172694295242339, 4.16533432858942224306190877309, 4.75194794536386495813638676141, 5.30343333664926211319715971269, 5.95264744156061957621713588411, 6.16947576792430642703492773079, 6.63142311653263516132064911194, 7.15881746019521794633840806448, 7.69948647082279765704915061134, 7.941271392306644231221917118094, 8.435841433342748560325439996034, 8.765441500504530385676685487280, 9.108091983481405883066704444257, 9.147582540727085318460012811390, 10.21562198516954790051202097404
