| L(s) = 1 | + 3-s + 4-s + 4·7-s + 9-s + 12-s − 8·13-s + 16-s + 2·19-s + 4·21-s + 25-s + 27-s + 4·28-s + 16·31-s + 36-s + 16·37-s − 8·39-s + 4·43-s + 48-s − 2·49-s − 8·52-s + 2·57-s + 4·61-s + 4·63-s + 64-s − 32·67-s − 20·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 0.458·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.755·28-s + 2.87·31-s + 1/6·36-s + 2.63·37-s − 1.28·39-s + 0.609·43-s + 0.144·48-s − 2/7·49-s − 1.10·52-s + 0.264·57-s + 0.512·61-s + 0.503·63-s + 1/8·64-s − 3.90·67-s − 2.34·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 974700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.575420080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575420080\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.025411245427374533367092169321, −7.71815239727995297209637936624, −7.40324461827821970052106158768, −7.05793241081314730223457125124, −6.31332901102351283696951953945, −6.02018801955886064007060925313, −5.34546022558899141381965141097, −4.69454483129358392143517429094, −4.56694930325129271519942535650, −4.28650910220876727399058644882, −2.97710380529728573896163389264, −2.91133319867084723778855091440, −2.30283657246697061436212829968, −1.66366453671526298770292201583, −0.879125982758298393177104864028,
0.879125982758298393177104864028, 1.66366453671526298770292201583, 2.30283657246697061436212829968, 2.91133319867084723778855091440, 2.97710380529728573896163389264, 4.28650910220876727399058644882, 4.56694930325129271519942535650, 4.69454483129358392143517429094, 5.34546022558899141381965141097, 6.02018801955886064007060925313, 6.31332901102351283696951953945, 7.05793241081314730223457125124, 7.40324461827821970052106158768, 7.71815239727995297209637936624, 8.025411245427374533367092169321
