| L(s) = 1 | − 5-s − 2·7-s + 3·11-s + 4·13-s − 12·17-s − 14·19-s + 6·23-s + 3·29-s − 5·31-s + 2·35-s − 8·37-s + 3·41-s − 8·43-s + 7·49-s + 12·53-s − 3·55-s − 3·59-s − 14·61-s − 4·65-s − 2·67-s − 30·71-s − 20·73-s − 6·77-s − 8·79-s + 12·85-s − 30·89-s − 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s + 1.10·13-s − 2.91·17-s − 3.21·19-s + 1.25·23-s + 0.557·29-s − 0.898·31-s + 0.338·35-s − 1.31·37-s + 0.468·41-s − 1.21·43-s + 49-s + 1.64·53-s − 0.404·55-s − 0.390·59-s − 1.79·61-s − 0.496·65-s − 0.244·67-s − 3.56·71-s − 2.34·73-s − 0.683·77-s − 0.900·79-s + 1.30·85-s − 3.17·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2770568995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2770568995\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.388187937159574100785106310293, −8.863952412991621595377410101591, −8.782409712783191266003475107892, −8.583795628938584984925678646112, −8.453013152085747049787730787258, −7.22126982589462017211327122472, −7.17175480690019950376715801518, −6.86970356997499493542957910144, −6.34169824154283675164755073432, −6.01723067690882447761689464330, −5.85606818802726019282396319318, −4.71624765719265718780979567555, −4.55282514854067970543481622296, −4.06077036208961075527255749196, −3.94269624297827293749372943913, −3.05430964810839070393835231660, −2.72049172470732676531179563086, −1.88065427957924262752637930011, −1.57282922326437926704894201169, −0.19473956684164253545383490904,
0.19473956684164253545383490904, 1.57282922326437926704894201169, 1.88065427957924262752637930011, 2.72049172470732676531179563086, 3.05430964810839070393835231660, 3.94269624297827293749372943913, 4.06077036208961075527255749196, 4.55282514854067970543481622296, 4.71624765719265718780979567555, 5.85606818802726019282396319318, 6.01723067690882447761689464330, 6.34169824154283675164755073432, 6.86970356997499493542957910144, 7.17175480690019950376715801518, 7.22126982589462017211327122472, 8.453013152085747049787730787258, 8.583795628938584984925678646112, 8.782409712783191266003475107892, 8.863952412991621595377410101591, 9.388187937159574100785106310293
