| L(s) = 1 | − 3·4-s − 10·13-s + 5·16-s − 4·19-s + 12·31-s + 10·37-s + 20·43-s − 14·49-s + 30·52-s − 22·61-s − 3·64-s + 20·73-s + 12·76-s + 24·79-s + 10·97-s − 20·103-s + 4·109-s + 3·121-s − 36·124-s + 127-s + 131-s + 137-s + 139-s − 30·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.77·13-s + 5/4·16-s − 0.917·19-s + 2.15·31-s + 1.64·37-s + 3.04·43-s − 2·49-s + 4.16·52-s − 2.81·61-s − 3/8·64-s + 2.34·73-s + 1.37·76-s + 2.70·79-s + 1.01·97-s − 1.97·103-s + 0.383·109-s + 3/11·121-s − 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.46·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−8.111291554543981630243615144391, −8.045590620524125419129335461921, −7.61064821647543776585659150038, −7.09305325128470420948222804467, −6.24877777925675414442975442826, −6.20481852631863748855544323115, −5.27951774771441848148740620672, −4.72067276749777115352779746535, −4.69824702877576535283262083774, −4.23023899623652582277153417738, −3.45045562084777234969796191815, −2.52397037751510802984424134455, −2.40058573044279568120832059087, −0.929391360715818373685365786299, 0,
0.929391360715818373685365786299, 2.40058573044279568120832059087, 2.52397037751510802984424134455, 3.45045562084777234969796191815, 4.23023899623652582277153417738, 4.69824702877576535283262083774, 4.72067276749777115352779746535, 5.27951774771441848148740620672, 6.20481852631863748855544323115, 6.24877777925675414442975442826, 7.09305325128470420948222804467, 7.61064821647543776585659150038, 8.045590620524125419129335461921, 8.111291554543981630243615144391
