| L(s) = 1 | − 4·4-s + 12·16-s − 12·23-s + 2·25-s − 12·29-s + 2·43-s − 11·49-s − 24·53-s + 2·61-s − 32·64-s − 22·79-s + 48·92-s − 8·100-s − 36·101-s + 2·103-s + 12·107-s − 12·113-s + 48·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 2.50·23-s + 2/5·25-s − 2.22·29-s + 0.304·43-s − 1.57·49-s − 3.29·53-s + 0.256·61-s − 4·64-s − 2.47·79-s + 5.00·92-s − 4/5·100-s − 3.58·101-s + 0.197·103-s + 1.16·107-s − 1.12·113-s + 4.45·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−9.263350720538986503949105687311, −9.023759865092393766925130882852, −8.433072077157405642270976195742, −8.050875138516453875748684461194, −7.86295233529838917236644390129, −7.56773026814495758560850966451, −6.78191667369722655303971926722, −6.31271200054070109359894624843, −5.78042354082766657034611225905, −5.58955418777624952506230146765, −5.04981090419579758935387031604, −4.62565579207585428587006442781, −4.01314425454951976637895234458, −4.00306439752385457084057769673, −3.34122237006366957014266780101, −2.82072525284652420726382253881, −1.77572593776434704193686502476, −1.41433462402830073203810317989, 0, 0,
1.41433462402830073203810317989, 1.77572593776434704193686502476, 2.82072525284652420726382253881, 3.34122237006366957014266780101, 4.00306439752385457084057769673, 4.01314425454951976637895234458, 4.62565579207585428587006442781, 5.04981090419579758935387031604, 5.58955418777624952506230146765, 5.78042354082766657034611225905, 6.31271200054070109359894624843, 6.78191667369722655303971926722, 7.56773026814495758560850966451, 7.86295233529838917236644390129, 8.050875138516453875748684461194, 8.433072077157405642270976195742, 9.023759865092393766925130882852, 9.263350720538986503949105687311
