| L(s) = 1 | − 4·7-s − 4·17-s + 12·23-s − 8·25-s − 16·31-s + 16·47-s − 2·49-s + 20·71-s + 8·73-s − 8·89-s − 4·97-s + 12·103-s − 12·113-s + 16·119-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 48·161-s + 163-s + 167-s − 24·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.970·17-s + 2.50·23-s − 8/5·25-s − 2.87·31-s + 2.33·47-s − 2/7·49-s + 2.37·71-s + 0.936·73-s − 0.847·89-s − 0.406·97-s + 1.18·103-s − 1.12·113-s + 1.46·119-s − 1.81·121-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 − 3.78·161-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.744796432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744796432\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 84 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−7.63438919338293602430347308191, −7.52744505199193941504454716700, −7.20225080761058324462132967572, −6.81661444948021737503267202348, −6.56198873272250189527218510595, −6.33867457918694897919168326919, −5.75223234913966853225051170812, −5.55508660314495972686547193038, −5.16703740086058051182335791033, −4.92691908064845991080776551356, −4.15550209911613096501062135269, −4.06268716982881348786700568684, −3.55309748625572094089636450214, −3.38240327711582271964943877296, −2.70050193025494423084594068380, −2.65307163020726127418128630730, −1.78506719443029873857560523738, −1.76897001394204915176789813349, −0.68930109206845918516204097731, −0.42721575514083964807241665263,
0.42721575514083964807241665263, 0.68930109206845918516204097731, 1.76897001394204915176789813349, 1.78506719443029873857560523738, 2.65307163020726127418128630730, 2.70050193025494423084594068380, 3.38240327711582271964943877296, 3.55309748625572094089636450214, 4.06268716982881348786700568684, 4.15550209911613096501062135269, 4.92691908064845991080776551356, 5.16703740086058051182335791033, 5.55508660314495972686547193038, 5.75223234913966853225051170812, 6.33867457918694897919168326919, 6.56198873272250189527218510595, 6.81661444948021737503267202348, 7.20225080761058324462132967572, 7.52744505199193941504454716700, 7.63438919338293602430347308191
