L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 3·9-s − 3·11-s − 2·13-s − 2·15-s − 7·17-s − 3·19-s − 4·21-s + 3·23-s + 5·25-s + 4·27-s + 9·29-s − 16·31-s − 6·33-s + 2·35-s − 37-s − 4·39-s − 2·41-s − 3·43-s − 3·45-s + 3·49-s − 14·51-s + 20·53-s + 3·55-s − 6·57-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 9-s − 0.904·11-s − 0.554·13-s − 0.516·15-s − 1.69·17-s − 0.688·19-s − 0.872·21-s + 0.625·23-s + 25-s + 0.769·27-s + 1.67·29-s − 2.87·31-s − 1.04·33-s + 0.338·35-s − 0.164·37-s − 0.640·39-s − 0.312·41-s − 0.457·43-s − 0.447·45-s + 3/7·49-s − 1.96·51-s + 2.74·53-s + 0.404·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19079424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + T + 60 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 126 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 118 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
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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
−8.216900614190127963971367518762, −7.889392026427067423410072936583, −7.40131674760284762396632088568, −7.07313971718622245341668269303, −6.87691475617728377043492692442, −6.68630345908112665857104382450, −5.85692606612607979901153052624, −5.73953933465557490166842826982, −5.10823540454177304436649670106, −4.65692390530241021993424807708, −4.31409506244711319978082290316, −4.13342882534706977989963386870, −3.41440056249570398766029959707, −3.04484818728912692464358973393, −2.77177238614580376965915239069, −2.38219605281060906851939569079, −1.81476262327988030858808894055, −1.26332050216584940928299380318, 0, 0,
1.26332050216584940928299380318, 1.81476262327988030858808894055, 2.38219605281060906851939569079, 2.77177238614580376965915239069, 3.04484818728912692464358973393, 3.41440056249570398766029959707, 4.13342882534706977989963386870, 4.31409506244711319978082290316, 4.65692390530241021993424807708, 5.10823540454177304436649670106, 5.73953933465557490166842826982, 5.85692606612607979901153052624, 6.68630345908112665857104382450, 6.87691475617728377043492692442, 7.07313971718622245341668269303, 7.40131674760284762396632088568, 7.889392026427067423410072936583, 8.216900614190127963971367518762