| L(s) = 1 | − 3-s − 2·9-s + 25-s + 5·27-s + 10·31-s + 10·37-s + 5·49-s − 8·67-s − 75-s + 81-s − 10·93-s − 12·97-s + 20·103-s − 10·111-s − 11·121-s + 127-s + 131-s + 137-s + 139-s − 5·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 1/5·25-s + 0.962·27-s + 1.79·31-s + 1.64·37-s + 5/7·49-s − 0.977·67-s − 0.115·75-s + 1/9·81-s − 1.03·93-s − 1.21·97-s + 1.97·103-s − 0.949·111-s − 121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.412·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.467877982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.467877982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 121 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.87593654303349886857172077080, −7.40089264174184854820220349249, −6.89959855892061815346803427919, −6.45490567529153224316041407688, −6.03503403089325110773932856576, −5.84640189028514672926902501059, −5.17000253331662304363673919147, −4.83410991664986751020192288084, −4.32685908007666435600559165157, −3.90046835191341102603958250324, −3.06508779363212207018932415185, −2.79552916680326475647353080209, −2.19428086878722268444521359470, −1.24420742243129361157814848906, −0.57657177492431182439495503410,
0.57657177492431182439495503410, 1.24420742243129361157814848906, 2.19428086878722268444521359470, 2.79552916680326475647353080209, 3.06508779363212207018932415185, 3.90046835191341102603958250324, 4.32685908007666435600559165157, 4.83410991664986751020192288084, 5.17000253331662304363673919147, 5.84640189028514672926902501059, 6.03503403089325110773932856576, 6.45490567529153224316041407688, 6.89959855892061815346803427919, 7.40089264174184854820220349249, 7.87593654303349886857172077080
