| L(s) = 1 | − 3-s + 4-s − 2·7-s + 9-s − 12-s + 4·13-s + 16-s − 16·19-s + 2·21-s − 6·25-s − 27-s − 2·28-s + 16·31-s + 36-s + 12·37-s − 4·39-s + 16·43-s − 48-s + 3·49-s + 4·52-s + 16·57-s − 28·61-s − 2·63-s + 64-s − 8·67-s − 28·73-s + 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 3.67·19-s + 0.436·21-s − 6/5·25-s − 0.192·27-s − 0.377·28-s + 2.87·31-s + 1/6·36-s + 1.97·37-s − 0.640·39-s + 2.43·43-s − 0.144·48-s + 3/7·49-s + 0.554·52-s + 2.11·57-s − 3.58·61-s − 0.251·63-s + 1/8·64-s − 0.977·67-s − 3.27·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640332 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.260375317206775288322444616854, −7.54842093789349707312948138248, −7.34608818311813371377464448078, −6.41309505638029156456085338806, −6.21868608096758870562967590194, −6.10611924951092571905001273028, −5.83815977597462037088554178569, −4.52496529634501691830241759811, −4.27546340147020629083452713682, −4.24578358603152871833997851063, −3.11903064072712609997880205813, −2.65895736335338182549695498478, −2.02661744514455988507183410743, −1.14483701940607571217981775775, 0,
1.14483701940607571217981775775, 2.02661744514455988507183410743, 2.65895736335338182549695498478, 3.11903064072712609997880205813, 4.24578358603152871833997851063, 4.27546340147020629083452713682, 4.52496529634501691830241759811, 5.83815977597462037088554178569, 6.10611924951092571905001273028, 6.21868608096758870562967590194, 6.41309505638029156456085338806, 7.34608818311813371377464448078, 7.54842093789349707312948138248, 8.260375317206775288322444616854
