| L(s) = 1 | + 7-s − 13-s + 19-s + 25-s − 31-s − 3·37-s + 2·43-s + 3·67-s − 73-s − 79-s − 91-s + 4·97-s + 3·103-s − 3·109-s − 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 175-s + ⋯ |
| L(s) = 1 | + 7-s − 13-s + 19-s + 25-s − 31-s − 3·37-s + 2·43-s + 3·67-s − 73-s − 79-s − 91-s + 4·97-s + 3·103-s − 3·109-s − 121-s + 127-s + 131-s + 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10732176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10732176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.512209267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512209267\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$ | \( ( 1 - T )^{4} \) |
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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
−8.808018708222388361083739690713, −8.721428239091977409328540396038, −8.401355427901258069745534586826, −7.66521178533855076083032342665, −7.45735289571565894306087454559, −7.43222892996541527102605340630, −6.75956202037514628747692063659, −6.56800741298661335828446199808, −5.88678262867663424321559775989, −5.44319723677949011773746802858, −5.15096420313049167300489309310, −4.97000157823228307008810694099, −4.44807774789521778769943661440, −3.97272916804872554483507145888, −3.39270829817011179627149256740, −3.16915981514255235502836625509, −2.39169030920415275690909324094, −2.05766945114930847803090718769, −1.51273442492794027975358946607, −0.76928232423461740911054470868,
0.76928232423461740911054470868, 1.51273442492794027975358946607, 2.05766945114930847803090718769, 2.39169030920415275690909324094, 3.16915981514255235502836625509, 3.39270829817011179627149256740, 3.97272916804872554483507145888, 4.44807774789521778769943661440, 4.97000157823228307008810694099, 5.15096420313049167300489309310, 5.44319723677949011773746802858, 5.88678262867663424321559775989, 6.56800741298661335828446199808, 6.75956202037514628747692063659, 7.43222892996541527102605340630, 7.45735289571565894306087454559, 7.66521178533855076083032342665, 8.401355427901258069745534586826, 8.721428239091977409328540396038, 8.808018708222388361083739690713
