L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 6·13-s + 3·19-s + 3·21-s + 7·25-s + 5·27-s + 31-s + 5·37-s + 6·39-s − 8·43-s + 2·49-s − 3·57-s + 22·61-s + 6·63-s + 5·67-s + 73-s − 7·75-s − 17·79-s + 81-s + 18·91-s − 93-s + 12·97-s − 103-s − 11·109-s − 5·111-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.66·13-s + 0.688·19-s + 0.654·21-s + 7/5·25-s + 0.962·27-s + 0.179·31-s + 0.821·37-s + 0.960·39-s − 1.21·43-s + 2/7·49-s − 0.397·57-s + 2.81·61-s + 0.755·63-s + 0.610·67-s + 0.117·73-s − 0.808·75-s − 1.91·79-s + 1/9·81-s + 1.88·91-s − 0.103·93-s + 1.21·97-s − 0.0985·103-s − 1.05·109-s − 0.474·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.345003189241069038425144259427, −7.891485977463956497611526334652, −7.29563128178746022055830342476, −6.83128814367668579028050956782, −6.63815064709444361317617534186, −6.03943810773537879093425896765, −5.45091988678237022791731347780, −5.10958658658066901021790818529, −4.69706026158527176731437600073, −3.92723299345378771904182979240, −3.23084496472940598599232750068, −2.79464447993339900578561612890, −2.28803542447379881403124909488, −0.970113145048901789709954203376, 0,
0.970113145048901789709954203376, 2.28803542447379881403124909488, 2.79464447993339900578561612890, 3.23084496472940598599232750068, 3.92723299345378771904182979240, 4.69706026158527176731437600073, 5.10958658658066901021790818529, 5.45091988678237022791731347780, 6.03943810773537879093425896765, 6.63815064709444361317617534186, 6.83128814367668579028050956782, 7.29563128178746022055830342476, 7.891485977463956497611526334652, 8.345003189241069038425144259427