| L(s) = 1 | + 2·5-s − 6·9-s + 4·17-s + 4·19-s + 3·25-s − 16·31-s − 12·45-s + 2·49-s − 8·59-s − 4·61-s + 16·67-s − 12·73-s + 27·81-s + 8·85-s + 8·95-s + 12·101-s + 8·103-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s − 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s + 0.970·17-s + 0.917·19-s + 3/5·25-s − 2.87·31-s − 1.78·45-s + 2/7·49-s − 1.04·59-s − 0.512·61-s + 1.95·67-s − 1.40·73-s + 3·81-s + 0.867·85-s + 0.820·95-s + 1.19·101-s + 0.788·103-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.84210102043042646384025401352, −7.46212672600372947009312463710, −6.97238544391138654785050907556, −6.37399344243820867996093621951, −5.86830391534173005162345630789, −5.70093888866808737514223874840, −5.20324548069731475827972490709, −5.02429913120480640214209205427, −4.05130010083233398855700190379, −3.32072280545665331707726425753, −3.26180587408034592610487247010, −2.47692318206012541987277725331, −1.96992891210281049758559614540, −1.13847461285225699369638878979, 0,
1.13847461285225699369638878979, 1.96992891210281049758559614540, 2.47692318206012541987277725331, 3.26180587408034592610487247010, 3.32072280545665331707726425753, 4.05130010083233398855700190379, 5.02429913120480640214209205427, 5.20324548069731475827972490709, 5.70093888866808737514223874840, 5.86830391534173005162345630789, 6.37399344243820867996093621951, 6.97238544391138654785050907556, 7.46212672600372947009312463710, 7.84210102043042646384025401352
