| L(s) = 1 | − 4·3-s + 4-s + 2·7-s + 6·9-s + 6·11-s − 4·12-s − 3·13-s − 3·16-s + 17-s − 8·21-s + 4·27-s + 2·28-s + 3·29-s − 12·31-s − 24·33-s + 6·36-s − 11·37-s + 12·39-s − 3·41-s − 8·43-s + 6·44-s − 14·47-s + 12·48-s − 6·49-s − 4·51-s − 3·52-s − 11·53-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1/2·4-s + 0.755·7-s + 2·9-s + 1.80·11-s − 1.15·12-s − 0.832·13-s − 3/4·16-s + 0.242·17-s − 1.74·21-s + 0.769·27-s + 0.377·28-s + 0.557·29-s − 2.15·31-s − 4.17·33-s + 36-s − 1.80·37-s + 1.92·39-s − 0.468·41-s − 1.21·43-s + 0.904·44-s − 2.04·47-s + 1.73·48-s − 6/7·49-s − 0.560·51-s − 0.416·52-s − 1.51·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8046711975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8046711975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 53 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 105 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 153 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 138 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 130 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T + 117 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 193 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.80669947915824863712391959216, −7.24160893893448324486650255273, −7.09367141551672118955297101100, −6.63464541691598933812168329144, −6.59047888941572921282542421178, −6.33470489218713852908434629411, −5.77772767210869708758353102335, −5.56820953714516473098882604382, −5.04516660089517054094465786514, −4.99896786249472197348797751905, −4.61253166921471552737494813986, −4.33269845928362546909768716229, −3.62532803910219176447882149536, −3.27653902451895067109061060331, −3.05634352567261780926335228498, −1.96507461885938257255111882315, −1.84740275996839805231171468919, −1.58402223836770699819465286519, −0.75153563844865592177481562054, −0.32162293321254812869381010003,
0.32162293321254812869381010003, 0.75153563844865592177481562054, 1.58402223836770699819465286519, 1.84740275996839805231171468919, 1.96507461885938257255111882315, 3.05634352567261780926335228498, 3.27653902451895067109061060331, 3.62532803910219176447882149536, 4.33269845928362546909768716229, 4.61253166921471552737494813986, 4.99896786249472197348797751905, 5.04516660089517054094465786514, 5.56820953714516473098882604382, 5.77772767210869708758353102335, 6.33470489218713852908434629411, 6.59047888941572921282542421178, 6.63464541691598933812168329144, 7.09367141551672118955297101100, 7.24160893893448324486650255273, 7.80669947915824863712391959216
