| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 9-s − 4·11-s + 14-s + 16-s − 18-s + 4·22-s − 12·23-s + 2·25-s − 28-s − 12·29-s − 32-s + 36-s − 4·37-s − 8·43-s − 4·44-s + 12·46-s + 49-s − 2·50-s + 12·53-s + 56-s + 12·58-s − 63-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.852·22-s − 2.50·23-s + 2/5·25-s − 0.188·28-s − 2.22·29-s − 0.176·32-s + 1/6·36-s − 0.657·37-s − 1.21·43-s − 0.603·44-s + 1.76·46-s + 1/7·49-s − 0.282·50-s + 1.64·53-s + 0.133·56-s + 1.57·58-s − 0.125·63-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49392 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.963683812138799647694117805840, −9.455233230626296297581383691014, −8.800075982019565868194983938988, −8.275905883481384063496399826650, −7.83666544657320129662626930537, −7.34169956228273066275321969691, −6.84358541519263684463629679116, −6.09000114680302489266057172235, −5.61674087945640100011160258562, −5.04771075019211386110222348154, −4.02028015831506240685693039629, −3.53634396972515340978525015086, −2.47781673599694344290126145508, −1.82418838464450996534852307739, 0,
1.82418838464450996534852307739, 2.47781673599694344290126145508, 3.53634396972515340978525015086, 4.02028015831506240685693039629, 5.04771075019211386110222348154, 5.61674087945640100011160258562, 6.09000114680302489266057172235, 6.84358541519263684463629679116, 7.34169956228273066275321969691, 7.83666544657320129662626930537, 8.275905883481384063496399826650, 8.800075982019565868194983938988, 9.455233230626296297581383691014, 9.963683812138799647694117805840
