| L(s) = 1 | + 3-s − 2·4-s + 3·7-s + 6·11-s − 2·12-s − 6·19-s + 3·21-s + 6·23-s − 2·25-s − 27-s − 6·28-s − 6·29-s + 6·33-s + 12·41-s + 43-s − 12·44-s − 49-s + 24·53-s − 6·57-s + 6·59-s − 61-s + 8·64-s − 15·67-s + 6·69-s + 18·71-s − 2·75-s + 12·76-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.13·7-s + 1.80·11-s − 0.577·12-s − 1.37·19-s + 0.654·21-s + 1.25·23-s − 2/5·25-s − 0.192·27-s − 1.13·28-s − 1.11·29-s + 1.04·33-s + 1.87·41-s + 0.152·43-s − 1.80·44-s − 1/7·49-s + 3.29·53-s − 0.794·57-s + 0.781·59-s − 0.128·61-s + 64-s − 1.83·67-s + 0.722·69-s + 2.13·71-s − 0.230·75-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.042959419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.042959419\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 89 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 179 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−11.36147182238873260024867362005, −10.49442440648619350114865378177, −10.41221220614766045981586302825, −9.406203806181377274711800818783, −9.347911968868255416253747207021, −8.862486955937312646303034722741, −8.656125750921008090335996092206, −8.144735834633521420420117327801, −7.64979674196873930357252209275, −6.89307824585006270304523573160, −6.86125411285924226070362454242, −5.78878743226053558966862848467, −5.65401343942211927856765120769, −4.76750174965205353128656366852, −4.23481505860246828027979729659, −4.13859799845170209925812116737, −3.44685754561145071373143351144, −2.45271418908805836296304860368, −1.82616157343769518731569615942, −0.906293462322824100179264880209,
0.906293462322824100179264880209, 1.82616157343769518731569615942, 2.45271418908805836296304860368, 3.44685754561145071373143351144, 4.13859799845170209925812116737, 4.23481505860246828027979729659, 4.76750174965205353128656366852, 5.65401343942211927856765120769, 5.78878743226053558966862848467, 6.86125411285924226070362454242, 6.89307824585006270304523573160, 7.64979674196873930357252209275, 8.144735834633521420420117327801, 8.656125750921008090335996092206, 8.862486955937312646303034722741, 9.347911968868255416253747207021, 9.406203806181377274711800818783, 10.41221220614766045981586302825, 10.49442440648619350114865378177, 11.36147182238873260024867362005
