| L(s) = 1 | + 3-s + 5-s − 2·9-s + 15-s − 4·16-s + 2·23-s − 4·25-s − 5·27-s + 14·31-s − 2·45-s + 16·47-s − 4·48-s − 10·49-s − 12·53-s + 2·69-s − 4·75-s − 4·80-s + 81-s + 14·93-s − 18·113-s + 2·115-s − 9·125-s + 127-s + 131-s − 5·135-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s + 0.258·15-s − 16-s + 0.417·23-s − 4/5·25-s − 0.962·27-s + 2.51·31-s − 0.298·45-s + 2.33·47-s − 0.577·48-s − 1.42·49-s − 1.64·53-s + 0.240·69-s − 0.461·75-s − 0.447·80-s + 1/9·81-s + 1.45·93-s − 1.69·113-s + 0.186·115-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s − 0.430·135-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.39893594642771643206543547031, −6.80044609023638042209450417099, −6.47231251560475429244143852591, −6.09966792552373038662002103453, −5.73870626081545475442824548628, −5.13615482341982682420909731232, −4.77016036292615901681122983244, −4.30413573577731527020603612370, −3.82123745760857088121774501968, −3.21993795263974952405435891110, −2.67401217228157830548156869283, −2.46478514923094428626030938829, −1.78907311686180992193673629119, −1.03757175929891373992477223583, 0,
1.03757175929891373992477223583, 1.78907311686180992193673629119, 2.46478514923094428626030938829, 2.67401217228157830548156869283, 3.21993795263974952405435891110, 3.82123745760857088121774501968, 4.30413573577731527020603612370, 4.77016036292615901681122983244, 5.13615482341982682420909731232, 5.73870626081545475442824548628, 6.09966792552373038662002103453, 6.47231251560475429244143852591, 6.80044609023638042209450417099, 7.39893594642771643206543547031
