| L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s + 9-s − 2·11-s + 2·14-s + 5·16-s − 2·18-s + 4·22-s − 8·23-s − 10·25-s − 3·28-s + 12·29-s − 6·32-s + 3·36-s + 20·37-s + 16·43-s − 6·44-s + 16·46-s + 49-s + 20·50-s − 20·53-s + 4·56-s − 24·58-s − 63-s + 7·64-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s − 0.603·11-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 0.852·22-s − 1.66·23-s − 2·25-s − 0.566·28-s + 2.22·29-s − 1.06·32-s + 1/2·36-s + 3.28·37-s + 2.43·43-s − 0.904·44-s + 2.35·46-s + 1/7·49-s + 2.82·50-s − 2.74·53-s + 0.534·56-s − 3.15·58-s − 0.125·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1494108 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7197813021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7197813021\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83922398202642458587198828500, −7.79089756844307556875120056782, −7.30932067156140291452614810498, −6.50358738357793944503631024403, −6.37119763678498512356952453412, −5.90660738221292552490908354487, −5.61053425475447892671224791397, −4.70213202876017617150319733849, −4.23003763111100218531402281021, −3.88277706681863956744577131696, −2.90909027996993607554406630510, −2.63621530561166833588250288036, −2.07103752121128576013331888677, −1.28532701621031142017445445957, −0.48386265623252023695541121788,
0.48386265623252023695541121788, 1.28532701621031142017445445957, 2.07103752121128576013331888677, 2.63621530561166833588250288036, 2.90909027996993607554406630510, 3.88277706681863956744577131696, 4.23003763111100218531402281021, 4.70213202876017617150319733849, 5.61053425475447892671224791397, 5.90660738221292552490908354487, 6.37119763678498512356952453412, 6.50358738357793944503631024403, 7.30932067156140291452614810498, 7.79089756844307556875120056782, 7.83922398202642458587198828500
