| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s + 2·10-s − 3·11-s − 6·12-s − 4·14-s − 2·15-s + 5·16-s − 7·17-s + 6·18-s − 3·19-s + 3·20-s + 4·21-s − 6·22-s − 3·23-s − 8·24-s + 5·25-s − 4·27-s − 6·28-s + 9·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.447·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s + 0.632·10-s − 0.904·11-s − 1.73·12-s − 1.06·14-s − 0.516·15-s + 5/4·16-s − 1.69·17-s + 1.41·18-s − 0.688·19-s + 0.670·20-s + 0.872·21-s − 1.27·22-s − 0.625·23-s − 1.63·24-s + 25-s − 0.769·27-s − 1.13·28-s + 1.67·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.344664553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344664553\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| 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 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 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
−7.68634971261731841361530345503, −7.66355750417991140697761149300, −7.29520075964807195407607261495, −6.75219427046077400445373489633, −6.45522411028721007232063303897, −6.45392935700676377918815887291, −5.82210436202654263168435037062, −5.81326612013499965031562730859, −5.20828170826158105598078173925, −4.98440869850844187168806825124, −4.64435039634744777431865735220, −4.22906947203251665901274850447, −3.93396769669934073899606104063, −3.51738931187456474759756935512, −2.79288000868637548437647632133, −2.72574481248970376630769618616, −2.07170418726121768412641619059, −1.82285674105995606144252322200, −1.05970542514761891794884568006, −0.24003321774572802507874822758,
0.24003321774572802507874822758, 1.05970542514761891794884568006, 1.82285674105995606144252322200, 2.07170418726121768412641619059, 2.72574481248970376630769618616, 2.79288000868637548437647632133, 3.51738931187456474759756935512, 3.93396769669934073899606104063, 4.22906947203251665901274850447, 4.64435039634744777431865735220, 4.98440869850844187168806825124, 5.20828170826158105598078173925, 5.81326612013499965031562730859, 5.82210436202654263168435037062, 6.45392935700676377918815887291, 6.45522411028721007232063303897, 6.75219427046077400445373489633, 7.29520075964807195407607261495, 7.66355750417991140697761149300, 7.68634971261731841361530345503
