| L(s) = 1 | − 3-s − 5-s + 7-s + 11-s + 15-s − 21-s + 25-s + 27-s + 2·29-s − 31-s − 33-s − 35-s − 53-s − 55-s + 59-s − 2·73-s − 75-s + 77-s − 79-s − 81-s − 2·83-s − 2·87-s + 93-s − 2·97-s + 2·101-s + 2·103-s + 105-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s + 7-s + 11-s + 15-s − 21-s + 25-s + 27-s + 2·29-s − 31-s − 33-s − 35-s − 53-s − 55-s + 59-s − 2·73-s − 75-s + 77-s − 79-s − 81-s − 2·83-s − 2·87-s + 93-s − 2·97-s + 2·101-s + 2·103-s + 105-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5748560791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5748560791\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−11.07881001987800070929954829119, −10.70073126480046450237190066788, −10.08841910434541675615845718302, −9.883583031263375174942489316694, −8.933107251918218814393240957626, −8.802826805526790771060949719812, −8.328915675856932775456466862810, −7.982592131622134851783817364615, −7.22706844495125827639929926091, −7.08589775662895294288499797197, −6.48932399175406306709175052839, −6.00106889817306809408986297823, −5.53234022831618700437095351706, −4.92983018325448709752031177170, −4.43812409956674834889354206479, −4.28444397760405380763367379383, −3.35025979848401591248573946837, −2.90533663940179130952613923919, −1.80434783672156620923385269613, −0.997713551704961223929152415114,
0.997713551704961223929152415114, 1.80434783672156620923385269613, 2.90533663940179130952613923919, 3.35025979848401591248573946837, 4.28444397760405380763367379383, 4.43812409956674834889354206479, 4.92983018325448709752031177170, 5.53234022831618700437095351706, 6.00106889817306809408986297823, 6.48932399175406306709175052839, 7.08589775662895294288499797197, 7.22706844495125827639929926091, 7.982592131622134851783817364615, 8.328915675856932775456466862810, 8.802826805526790771060949719812, 8.933107251918218814393240957626, 9.883583031263375174942489316694, 10.08841910434541675615845718302, 10.70073126480046450237190066788, 11.07881001987800070929954829119
