| L(s) = 1 | − 2·2-s − 4·3-s + 3·4-s + 8·6-s − 2·7-s − 4·8-s + 6·9-s − 2·11-s − 12·12-s − 2·13-s + 4·14-s + 5·16-s + 2·17-s − 12·18-s − 2·19-s + 8·21-s + 4·22-s + 2·23-s + 16·24-s + 4·26-s + 4·27-s − 6·28-s + 6·29-s − 2·31-s − 6·32-s + 8·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 3/2·4-s + 3.26·6-s − 0.755·7-s − 1.41·8-s + 2·9-s − 0.603·11-s − 3.46·12-s − 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.485·17-s − 2.82·18-s − 0.458·19-s + 1.74·21-s + 0.852·22-s + 0.417·23-s + 3.26·24-s + 0.784·26-s + 0.769·27-s − 1.13·28-s + 1.11·29-s − 0.359·31-s − 1.06·32-s + 1.39·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 22 T + 282 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
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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
−8.219258908242235018153596843961, −8.130014966227299819415813346295, −7.27217043039939613556152074881, −7.26983555950026925281863795615, −6.69627114659404053778147701216, −6.63515838341925719101763811778, −6.01987031901699954339518435262, −5.87122519362157433728008435182, −5.50369414958207086046131236699, −5.10939266257305154357538387914, −4.68174321122328254398782262910, −4.32367678329535912781877119078, −3.42421131782095103206799609304, −3.17165809557258956893501301044, −2.43806178000908702264492520940, −2.26677785319632327953878773890, −1.06979406516936859324046319691, −0.997038494175514044118552061593, 0, 0,
0.997038494175514044118552061593, 1.06979406516936859324046319691, 2.26677785319632327953878773890, 2.43806178000908702264492520940, 3.17165809557258956893501301044, 3.42421131782095103206799609304, 4.32367678329535912781877119078, 4.68174321122328254398782262910, 5.10939266257305154357538387914, 5.50369414958207086046131236699, 5.87122519362157433728008435182, 6.01987031901699954339518435262, 6.63515838341925719101763811778, 6.69627114659404053778147701216, 7.26983555950026925281863795615, 7.27217043039939613556152074881, 8.130014966227299819415813346295, 8.219258908242235018153596843961
