| L(s) = 1 | − 3-s − 4·5-s + 9-s + 4·15-s − 8·19-s + 6·25-s − 27-s + 12·29-s − 8·43-s − 4·45-s + 16·47-s − 2·49-s − 4·53-s + 8·57-s + 8·67-s − 4·73-s − 6·75-s + 81-s − 12·87-s + 32·95-s + 4·97-s + 12·101-s + 10·121-s − 4·125-s + 127-s + 8·129-s + 131-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s + 1/3·9-s + 1.03·15-s − 1.83·19-s + 6/5·25-s − 0.192·27-s + 2.22·29-s − 1.21·43-s − 0.596·45-s + 2.33·47-s − 2/7·49-s − 0.549·53-s + 1.05·57-s + 0.977·67-s − 0.468·73-s − 0.692·75-s + 1/9·81-s − 1.28·87-s + 3.28·95-s + 0.406·97-s + 1.19·101-s + 0.909·121-s − 0.357·125-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.334580286466769938869095137697, −7.927228858346324339237951328163, −7.47184539419697069244714981888, −7.01102700383348905036395737363, −6.46956845524518855788592957843, −6.24990012986292931848349844623, −5.50821028045941260632498910659, −4.79025209594313437944303375568, −4.52353109771654662086830780511, −4.02631035226127556414869722600, −3.58103221487799765618724063676, −2.83760885317223549422167444783, −2.10122930606302984424679070602, −0.933135093514415288573050382372, 0,
0.933135093514415288573050382372, 2.10122930606302984424679070602, 2.83760885317223549422167444783, 3.58103221487799765618724063676, 4.02631035226127556414869722600, 4.52353109771654662086830780511, 4.79025209594313437944303375568, 5.50821028045941260632498910659, 6.24990012986292931848349844623, 6.46956845524518855788592957843, 7.01102700383348905036395737363, 7.47184539419697069244714981888, 7.927228858346324339237951328163, 8.334580286466769938869095137697
