| L(s) = 1 | + 2-s + 2·3-s − 4-s − 4·5-s + 2·6-s − 3·8-s + 3·9-s − 4·10-s − 2·12-s − 8·15-s − 16-s − 12·17-s + 3·18-s − 19-s + 4·20-s − 6·24-s + 2·25-s + 4·27-s − 8·30-s + 16·31-s + 5·32-s − 12·34-s − 3·36-s − 38-s + 12·40-s − 12·45-s − 2·48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s − 1.06·8-s + 9-s − 1.26·10-s − 0.577·12-s − 2.06·15-s − 1/4·16-s − 2.91·17-s + 0.707·18-s − 0.229·19-s + 0.894·20-s − 1.22·24-s + 2/5·25-s + 0.769·27-s − 1.46·30-s + 2.87·31-s + 0.883·32-s − 2.05·34-s − 1/2·36-s − 0.162·38-s + 1.89·40-s − 1.78·45-s − 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080897756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080897756\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.238253561689615963456339949883, −7.921460871442440189837475718707, −7.39080902187563863728417650807, −6.68569466352490003812219927306, −6.46787589567348650226556085189, −6.10768231212675033255659362088, −5.05425533805193220116651473777, −4.56991321080597356703354311172, −4.37618629394185670620160142247, −4.15975883116735153333531420145, −3.44365518362789223021999993571, −3.04853305484647947479311614764, −2.53461503012656985065918963823, −1.75844151919507482866111198427, −0.38916689889120047822199203465,
0.38916689889120047822199203465, 1.75844151919507482866111198427, 2.53461503012656985065918963823, 3.04853305484647947479311614764, 3.44365518362789223021999993571, 4.15975883116735153333531420145, 4.37618629394185670620160142247, 4.56991321080597356703354311172, 5.05425533805193220116651473777, 6.10768231212675033255659362088, 6.46787589567348650226556085189, 6.68569466352490003812219927306, 7.39080902187563863728417650807, 7.921460871442440189837475718707, 8.238253561689615963456339949883
