| L(s) = 1 | − 4·11-s − 4·16-s + 10·19-s + 20·29-s − 6·31-s + 16·41-s + 5·49-s − 20·59-s + 14·61-s + 16·71-s − 24·101-s − 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 16·176-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 16-s + 2.29·19-s + 3.71·29-s − 1.07·31-s + 2.49·41-s + 5/7·49-s − 2.60·59-s + 1.79·61-s + 1.89·71-s − 2.38·101-s − 0.957·109-s − 0.909·121-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 + 1.92·169-s + 0.0760·173-s + 1.20·176-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311412189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311412189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−12.73460214203850642110915677933, −11.86762722550013622798387474922, −11.69505015980260670220668210359, −10.89657133502083882389003808700, −10.66342157981520354662097079042, −10.12504009504936354468855500211, −9.404709123413616119375007601467, −9.312389168373981811178736766399, −8.500271287471624913381000410191, −7.896509426078001001688218766754, −7.65577345363067892628210313770, −6.91289580859699725719709395531, −6.48321588953719438989821492717, −5.66114704444160969580464941270, −5.16340751709186156363692836860, −4.67811541254359204024988961648, −3.89658176012304504172302630337, −2.81947525310129657968705132067, −2.61412619845600714917427497301, −1.03508980367567642243416015717,
1.03508980367567642243416015717, 2.61412619845600714917427497301, 2.81947525310129657968705132067, 3.89658176012304504172302630337, 4.67811541254359204024988961648, 5.16340751709186156363692836860, 5.66114704444160969580464941270, 6.48321588953719438989821492717, 6.91289580859699725719709395531, 7.65577345363067892628210313770, 7.896509426078001001688218766754, 8.500271287471624913381000410191, 9.312389168373981811178736766399, 9.404709123413616119375007601467, 10.12504009504936354468855500211, 10.66342157981520354662097079042, 10.89657133502083882389003808700, 11.69505015980260670220668210359, 11.86762722550013622798387474922, 12.73460214203850642110915677933
