| L(s) = 1 | + 2·5-s − 9-s + 4·11-s − 12·19-s − 25-s − 12·29-s − 20·31-s − 12·41-s − 2·45-s − 49-s + 8·55-s − 24·59-s + 28·61-s − 4·71-s − 16·79-s + 81-s − 12·89-s − 24·95-s − 4·99-s − 12·101-s + 28·109-s − 10·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s + 1.20·11-s − 2.75·19-s − 1/5·25-s − 2.22·29-s − 3.59·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s + 1.07·55-s − 3.12·59-s + 3.58·61-s − 0.474·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s − 2.46·95-s − 0.402·99-s − 1.19·101-s + 2.68·109-s − 0.909·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3081361429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3081361429\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.077982012850391101525928114207, −8.448031672138903000288655699172, −8.274512137559448455460535201643, −7.55628711068129622695664873114, −7.31629850307708500964429993834, −6.85000642018443407104843059337, −6.57160460943585065057136817172, −6.17629148876581429129953919448, −5.79431844707338327410518512145, −5.37476953136403275462481248555, −5.29514266980052317167154322762, −4.37898966101054919940999309713, −4.18834287360190066692406421460, −3.63499789754054518610993517307, −3.51718245214933408380273794475, −2.69092832449966498426826711707, −2.01100176092952639044898944376, −1.78303011523515690794133173816, −1.59462033990013488894225854547, −0.15157138060999760872668201614,
0.15157138060999760872668201614, 1.59462033990013488894225854547, 1.78303011523515690794133173816, 2.01100176092952639044898944376, 2.69092832449966498426826711707, 3.51718245214933408380273794475, 3.63499789754054518610993517307, 4.18834287360190066692406421460, 4.37898966101054919940999309713, 5.29514266980052317167154322762, 5.37476953136403275462481248555, 5.79431844707338327410518512145, 6.17629148876581429129953919448, 6.57160460943585065057136817172, 6.85000642018443407104843059337, 7.31629850307708500964429993834, 7.55628711068129622695664873114, 8.274512137559448455460535201643, 8.448031672138903000288655699172, 9.077982012850391101525928114207
