| L(s) = 1 | + 7-s − 19-s + 25-s + 31-s + 2·37-s − 3·61-s + 3·73-s − 2·103-s + 109-s + 121-s + 127-s + 131-s − 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 + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 7-s − 19-s + 25-s + 31-s + 2·37-s − 3·61-s + 3·73-s − 2·103-s + 109-s + 121-s + 127-s + 131-s − 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 + 191-s + 193-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.614314421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614314421\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 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
−9.135659301562618735321249153028, −8.618247523596536959582917030194, −8.144333217560944768817681313838, −8.132481934834223351429644026848, −7.75908394735487107800435273534, −7.24240878866382176936342261483, −6.65933011439478906344406943758, −6.65011018854591101526918286977, −5.96132422405039532395787407618, −5.78574191043008658946576078931, −5.19988151621924944622844172264, −4.67519170620335696976850862735, −4.43164244411343096016676482497, −4.31531262534358539178751567404, −3.38505386125645388290574646598, −3.16180271198753585258068653366, −2.42809693206574242714868529610, −2.15086398858067224343662859813, −1.43642449049563159275036196451, −0.854390528594128567359266792805,
0.854390528594128567359266792805, 1.43642449049563159275036196451, 2.15086398858067224343662859813, 2.42809693206574242714868529610, 3.16180271198753585258068653366, 3.38505386125645388290574646598, 4.31531262534358539178751567404, 4.43164244411343096016676482497, 4.67519170620335696976850862735, 5.19988151621924944622844172264, 5.78574191043008658946576078931, 5.96132422405039532395787407618, 6.65011018854591101526918286977, 6.65933011439478906344406943758, 7.24240878866382176936342261483, 7.75908394735487107800435273534, 8.132481934834223351429644026848, 8.144333217560944768817681313838, 8.618247523596536959582917030194, 9.135659301562618735321249153028
