| L(s) = 1 | − 4-s + 2·7-s + 3·13-s + 25-s − 2·28-s + 43-s + 49-s − 3·52-s − 61-s + 64-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 100-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s − 172-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s + 3·13-s + 25-s − 2·28-s + 43-s + 49-s − 3·52-s − 61-s + 64-s − 3·67-s + 73-s + 3·79-s + 6·91-s − 100-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s − 172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10556001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10556001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.796279405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.796279405\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−8.840072184102666070232218288755, −8.773531421980081763754388289357, −8.164575142244402909451989518430, −8.115951285452561963505383518771, −7.77305045492050339229067379153, −7.28020870363842873542245277028, −6.66969113749752101259928434276, −6.33087742920474596172874830226, −6.00817633412880593672152776205, −5.53478311431452852529617122078, −5.09407706400131017447352677758, −4.81180736765956577315101229411, −4.33931518699122910180995037902, −4.13738257279103385877408229951, −3.54018624267618144778190584879, −3.27224063039232803522736910679, −2.46967940259246759521080510718, −1.83043379647876163264005803160, −1.27923637187373894372943988321, −1.02962987883543418812191685697,
1.02962987883543418812191685697, 1.27923637187373894372943988321, 1.83043379647876163264005803160, 2.46967940259246759521080510718, 3.27224063039232803522736910679, 3.54018624267618144778190584879, 4.13738257279103385877408229951, 4.33931518699122910180995037902, 4.81180736765956577315101229411, 5.09407706400131017447352677758, 5.53478311431452852529617122078, 6.00817633412880593672152776205, 6.33087742920474596172874830226, 6.66969113749752101259928434276, 7.28020870363842873542245277028, 7.77305045492050339229067379153, 8.115951285452561963505383518771, 8.164575142244402909451989518430, 8.773531421980081763754388289357, 8.840072184102666070232218288755
