| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s − 12·11-s + 12-s + 10·13-s + 16-s + 2·18-s + 12·22-s + 6·23-s − 24-s − 10·25-s − 10·26-s − 5·27-s − 32-s − 12·33-s − 2·36-s + 4·37-s + 10·39-s − 12·44-s − 6·46-s + 48-s − 13·49-s + 10·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 3.61·11-s + 0.288·12-s + 2.77·13-s + 1/4·16-s + 0.471·18-s + 2.55·22-s + 1.25·23-s − 0.204·24-s − 2·25-s − 1.96·26-s − 0.962·27-s − 0.176·32-s − 2.08·33-s − 1/3·36-s + 0.657·37-s + 1.60·39-s − 1.80·44-s − 0.884·46-s + 0.144·48-s − 1.85·49-s + 1.41·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.111181314911992176212142931936, −8.684413720393242419277779815238, −8.297610084688077297742296971210, −7.87841813427368117647214734878, −7.73591843742988285704805588989, −6.98043076884761860510728876245, −6.02372043714400021098795022997, −5.81027585652647529822118508915, −5.38865185009354292107305454716, −4.54275466905147962254344391783, −3.59163897364434449988676374845, −2.99432139105097413830284496966, −2.60234956115962506172119076503, −1.59940757777111675726084768164, 0,
1.59940757777111675726084768164, 2.60234956115962506172119076503, 2.99432139105097413830284496966, 3.59163897364434449988676374845, 4.54275466905147962254344391783, 5.38865185009354292107305454716, 5.81027585652647529822118508915, 6.02372043714400021098795022997, 6.98043076884761860510728876245, 7.73591843742988285704805588989, 7.87841813427368117647214734878, 8.297610084688077297742296971210, 8.684413720393242419277779815238, 9.111181314911992176212142931936
