L(s) = 1 | − 2·3-s + 4-s + 4·7-s + 9-s − 6·11-s − 2·12-s + 16-s − 8·21-s − 25-s + 4·27-s + 4·28-s + 12·33-s + 36-s − 7·37-s − 6·41-s − 6·44-s + 12·47-s − 2·48-s − 2·49-s − 6·53-s + 4·63-s + 64-s + 22·67-s + 6·71-s − 5·73-s + 2·75-s − 24·77-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1.51·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 1/4·16-s − 1.74·21-s − 1/5·25-s + 0.769·27-s + 0.755·28-s + 2.08·33-s + 1/6·36-s − 1.15·37-s − 0.937·41-s − 0.904·44-s + 1.75·47-s − 0.288·48-s − 2/7·49-s − 0.824·53-s + 0.503·63-s + 1/8·64-s + 2.68·67-s + 0.712·71-s − 0.585·73-s + 0.230·75-s − 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−8.243066600691440244092796408545, −8.024778868112728340255808777545, −7.43446377146260621256736619591, −6.87635874022665005722103098989, −6.63472473552404030802376550930, −5.78914951182907135690218862747, −5.42959586695453850994387325018, −5.22756226686314992047744145161, −4.74950611326412960359631408397, −4.16865034978492496691497877748, −3.31732055197170722076093780832, −2.60769693733909896055969422271, −2.03060370630186558587181705117, −1.23099204999884115095589342028, 0,
1.23099204999884115095589342028, 2.03060370630186558587181705117, 2.60769693733909896055969422271, 3.31732055197170722076093780832, 4.16865034978492496691497877748, 4.74950611326412960359631408397, 5.22756226686314992047744145161, 5.42959586695453850994387325018, 5.78914951182907135690218862747, 6.63472473552404030802376550930, 6.87635874022665005722103098989, 7.43446377146260621256736619591, 8.024778868112728340255808777545, 8.243066600691440244092796408545