| L(s) = 1 | − 4·7-s − 3·9-s − 20·37-s + 16·43-s + 9·49-s + 12·63-s + 32·67-s − 8·79-s + 9·81-s − 4·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 9-s − 3.28·37-s + 2.43·43-s + 9/7·49-s + 1.51·63-s + 3.90·67-s − 0.900·79-s + 81-s − 0.383·109-s + 2·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7722745061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7722745061\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−9.140359473819741912045290429027, −9.005568796603216054148463994725, −8.576018554529311497264334828884, −8.286216706435420671037694790869, −7.69583774122624360742788065592, −7.29920281428733611690020109441, −6.93012124686086045475158498470, −6.41733731262631075275075007114, −6.36440104201510944507106353488, −5.67214574559406365429699901661, −5.31397458398503663343686473893, −5.17439977888095293148851471069, −4.27355983583208869261111283968, −3.90679468927351554000943454559, −3.30543849513860444509813513783, −3.25550939055484864900328375471, −2.37845823048201808574836704749, −2.23512148930571499620864151192, −1.16006993820417020799848605996, −0.33265120961226415085162446587,
0.33265120961226415085162446587, 1.16006993820417020799848605996, 2.23512148930571499620864151192, 2.37845823048201808574836704749, 3.25550939055484864900328375471, 3.30543849513860444509813513783, 3.90679468927351554000943454559, 4.27355983583208869261111283968, 5.17439977888095293148851471069, 5.31397458398503663343686473893, 5.67214574559406365429699901661, 6.36440104201510944507106353488, 6.41733731262631075275075007114, 6.93012124686086045475158498470, 7.29920281428733611690020109441, 7.69583774122624360742788065592, 8.286216706435420671037694790869, 8.576018554529311497264334828884, 9.005568796603216054148463994725, 9.140359473819741912045290429027
