| L(s) = 1 | − 3·3-s − 3·5-s + 6·9-s − 6·11-s − 5·13-s + 9·15-s + 5·17-s + 3·19-s − 8·23-s + 5·25-s − 9·27-s − 18·29-s + 7·31-s + 18·33-s + 5·37-s + 15·39-s − 6·41-s + 8·43-s − 18·45-s + 47-s + 7·49-s − 15·51-s + 20·53-s + 18·55-s − 9·57-s − 18·59-s + 6·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s + 2·9-s − 1.80·11-s − 1.38·13-s + 2.32·15-s + 1.21·17-s + 0.688·19-s − 1.66·23-s + 25-s − 1.73·27-s − 3.34·29-s + 1.25·31-s + 3.13·33-s + 0.821·37-s + 2.40·39-s − 0.937·41-s + 1.21·43-s − 2.68·45-s + 0.145·47-s + 49-s − 2.10·51-s + 2.74·53-s + 2.42·55-s − 1.19·57-s − 2.34·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.21588692221203179500467666783, −9.616774568013260844088834770050, −9.157743022436732513407789051753, −8.434824960489590997617174678118, −7.83452805084155089333762493145, −7.62856631743935795062380073440, −7.30418638627036425691365154569, −7.22625770743461330208687199847, −6.26223092524026538078556517028, −5.63002330909251917615298605546, −5.58050928132307884244095569482, −5.25104962021962065413101676039, −4.38664074109845751692807810484, −4.30958139307583089520083666362, −3.64115797832701012179039388111, −2.86962857196961273102971430762, −2.28803540315174946141778946955, −1.22295559690610853878416816173, 0, 0,
1.22295559690610853878416816173, 2.28803540315174946141778946955, 2.86962857196961273102971430762, 3.64115797832701012179039388111, 4.30958139307583089520083666362, 4.38664074109845751692807810484, 5.25104962021962065413101676039, 5.58050928132307884244095569482, 5.63002330909251917615298605546, 6.26223092524026538078556517028, 7.22625770743461330208687199847, 7.30418638627036425691365154569, 7.62856631743935795062380073440, 7.83452805084155089333762493145, 8.434824960489590997617174678118, 9.157743022436732513407789051753, 9.616774568013260844088834770050, 10.21588692221203179500467666783
