| L(s) = 1 | − 7-s − 3·19-s − 25-s + 3·31-s + 2·37-s − 2·43-s − 3·61-s + 2·67-s − 3·73-s + 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 7-s − 3·19-s − 25-s + 3·31-s + 2·37-s − 2·43-s − 3·61-s + 2·67-s − 3·73-s + 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7895972654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7895972654\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.054177946641286646910082250703, −8.648945698951114820520293425504, −8.362177901967644800640795167687, −7.965417739014474246010891126742, −7.80051716747258995739023934599, −7.15165673939126676052198142380, −6.49293798176864093981901029232, −6.44975179771677594213385084891, −6.33048902036013736782946845390, −5.85022465390939629681901195272, −5.29546437456800243625744192685, −4.49439022163802085464188620348, −4.48827454076173295228232081585, −4.20535541113172442382398115999, −3.45997084599980878527322066409, −3.02922798616085085357130631764, −2.63365814456375771932375669693, −2.09677632069388811530758308037, −1.57661444517640536435648763308, −0.53659147058404712339798999000,
0.53659147058404712339798999000, 1.57661444517640536435648763308, 2.09677632069388811530758308037, 2.63365814456375771932375669693, 3.02922798616085085357130631764, 3.45997084599980878527322066409, 4.20535541113172442382398115999, 4.48827454076173295228232081585, 4.49439022163802085464188620348, 5.29546437456800243625744192685, 5.85022465390939629681901195272, 6.33048902036013736782946845390, 6.44975179771677594213385084891, 6.49293798176864093981901029232, 7.15165673939126676052198142380, 7.80051716747258995739023934599, 7.965417739014474246010891126742, 8.362177901967644800640795167687, 8.648945698951114820520293425504, 9.054177946641286646910082250703
