| L(s) = 1 | − 3-s − 4·4-s − 4·7-s − 2·9-s + 4·12-s − 2·13-s + 12·16-s + 4·19-s + 4·21-s − 9·25-s + 5·27-s + 16·28-s − 6·31-s + 8·36-s − 22·37-s + 2·39-s − 8·43-s − 12·48-s − 2·49-s + 8·52-s − 4·57-s − 4·61-s + 8·63-s − 32·64-s − 2·67-s − 32·73-s + 9·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2·4-s − 1.51·7-s − 2/3·9-s + 1.15·12-s − 0.554·13-s + 3·16-s + 0.917·19-s + 0.872·21-s − 9/5·25-s + 0.962·27-s + 3.02·28-s − 1.07·31-s + 4/3·36-s − 3.61·37-s + 0.320·39-s − 1.21·43-s − 1.73·48-s − 2/7·49-s + 1.10·52-s − 0.529·57-s − 0.512·61-s + 1.00·63-s − 4·64-s − 0.244·67-s − 3.74·73-s + 1.03·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−8.585957272776804427928347593039, −8.528601897910997907766991611046, −7.76116009742030896542487448970, −7.21661224338768342466420756206, −6.70861289370917133535141316633, −5.82135849339387357739621546932, −5.76971326436299567396516934204, −5.13164450675750106555447729059, −4.75768041533452306802121891710, −3.93044483272471932820766011024, −3.29304594675951575785139026233, −3.23573160090042234483516934690, −1.66131996401817111295419273715, 0, 0,
1.66131996401817111295419273715, 3.23573160090042234483516934690, 3.29304594675951575785139026233, 3.93044483272471932820766011024, 4.75768041533452306802121891710, 5.13164450675750106555447729059, 5.76971326436299567396516934204, 5.82135849339387357739621546932, 6.70861289370917133535141316633, 7.21661224338768342466420756206, 7.76116009742030896542487448970, 8.528601897910997907766991611046, 8.585957272776804427928347593039
