| L(s) = 1 | + 2-s − 8-s − 9-s + 13-s − 16-s − 17-s − 18-s + 26-s + 29-s − 34-s − 37-s + 41-s − 49-s + 2·53-s + 58-s + 61-s + 64-s + 72-s + 2·73-s − 74-s + 82-s − 2·89-s + 2·97-s − 98-s + 101-s − 104-s + 2·106-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s − 9-s + 13-s − 16-s − 17-s − 18-s + 26-s + 29-s − 34-s − 37-s + 41-s − 49-s + 2·53-s + 58-s + 61-s + 64-s + 72-s + 2·73-s − 74-s + 82-s − 2·89-s + 2·97-s − 98-s + 101-s − 104-s + 2·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.323838318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323838318\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.09606082358940032598558818559, −9.643459345138870173566842067214, −8.942944284633502120401481000426, −8.918542431343625099747325996942, −8.530919110704119651830607033475, −8.164929632129156754462801537614, −7.64824748595528663401383321329, −6.85488248620044246365419529149, −6.77549554023365768424715301641, −6.04964774932371898863144009287, −6.01969890404621571585186463490, −5.26697232213288397708490348288, −5.15847957699909673059212022035, −4.33823457292943185879583522982, −4.21909839307984958885684191902, −3.37975533014347964290218458648, −3.30123926838938206817027822151, −2.46938913889263881808019186782, −2.08772791411382502850709799067, −0.856459917117394552281502224461,
0.856459917117394552281502224461, 2.08772791411382502850709799067, 2.46938913889263881808019186782, 3.30123926838938206817027822151, 3.37975533014347964290218458648, 4.21909839307984958885684191902, 4.33823457292943185879583522982, 5.15847957699909673059212022035, 5.26697232213288397708490348288, 6.01969890404621571585186463490, 6.04964774932371898863144009287, 6.77549554023365768424715301641, 6.85488248620044246365419529149, 7.64824748595528663401383321329, 8.164929632129156754462801537614, 8.530919110704119651830607033475, 8.918542431343625099747325996942, 8.942944284633502120401481000426, 9.643459345138870173566842067214, 10.09606082358940032598558818559
