| L(s) = 1 | − 12·13-s − 32·31-s − 28·37-s − 16·43-s − 48·61-s + 32·67-s + 12·73-s − 20·97-s − 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3.32·13-s − 5.74·31-s − 4.60·37-s − 2.43·43-s − 6.14·61-s + 3.90·67-s + 1.40·73-s − 2.03·97-s − 1.57·103-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 + 0.0783·163-s + 0.0773·167-s + 5.53·169-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 + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1987917500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1987917500\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 4174 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 5678 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.71886339429833989282044577935, −6.48025309271415395226688903072, −6.41240524255394704551971617468, −5.67911986366971022479639617788, −5.64749653399400544413999505531, −5.43795792056993857163647974123, −5.34913479379908636252854943937, −5.03915728486839694532115419330, −5.00744720231320885275314015094, −4.76633131928587333509935764556, −4.66814011074200986772432783276, −3.97646098470530002719021982382, −3.84321166853135384625585458645, −3.73311768758589942754111583601, −3.64080422203276302653170258620, −3.10189365500027386821573188929, −3.05148912933651794720509682972, −2.60165661037880905373806591439, −2.54113440634900879469858532847, −1.84390203760456091503028715490, −1.77893586542349874046849229673, −1.69490034958348369695939731910, −1.58395789567019944546252019652, −0.29109505398892284315956895880, −0.18853676145586278586098266679,
0.18853676145586278586098266679, 0.29109505398892284315956895880, 1.58395789567019944546252019652, 1.69490034958348369695939731910, 1.77893586542349874046849229673, 1.84390203760456091503028715490, 2.54113440634900879469858532847, 2.60165661037880905373806591439, 3.05148912933651794720509682972, 3.10189365500027386821573188929, 3.64080422203276302653170258620, 3.73311768758589942754111583601, 3.84321166853135384625585458645, 3.97646098470530002719021982382, 4.66814011074200986772432783276, 4.76633131928587333509935764556, 5.00744720231320885275314015094, 5.03915728486839694532115419330, 5.34913479379908636252854943937, 5.43795792056993857163647974123, 5.64749653399400544413999505531, 5.67911986366971022479639617788, 6.41240524255394704551971617468, 6.48025309271415395226688903072, 6.71886339429833989282044577935
