| L(s) = 1 | − 3-s + 4-s − 7-s + 9-s − 12-s − 13-s − 3·16-s + 21-s − 6·25-s − 27-s − 28-s − 8·31-s + 36-s + 12·37-s + 39-s + 8·43-s + 3·48-s − 6·49-s − 52-s + 12·61-s − 63-s − 7·64-s + 4·73-s + 6·75-s + 81-s + 84-s + 91-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 0.377·7-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 3/4·16-s + 0.218·21-s − 6/5·25-s − 0.192·27-s − 0.188·28-s − 1.43·31-s + 1/6·36-s + 1.97·37-s + 0.160·39-s + 1.21·43-s + 0.433·48-s − 6/7·49-s − 0.138·52-s + 1.53·61-s − 0.125·63-s − 7/8·64-s + 0.468·73-s + 0.692·75-s + 1/9·81-s + 0.109·84-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2457 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6204090178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6204090178\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.86712142261172265095003988916, −12.57734678591887601772847316331, −11.65711565681209698953976937280, −11.32775537258475749001289496612, −10.82259124853772396204677861897, −9.895562633630313165325886248135, −9.538291305055480850556186888532, −8.706765597420159753229260194052, −7.67878695985149338439137309795, −7.22491173883806576538760882989, −6.32414320647087875011874975365, −5.80924366427092837393079362574, −4.76633374078783302992479082311, −3.77404669452369364554475368343, −2.31828391835134807113696350943,
2.31828391835134807113696350943, 3.77404669452369364554475368343, 4.76633374078783302992479082311, 5.80924366427092837393079362574, 6.32414320647087875011874975365, 7.22491173883806576538760882989, 7.67878695985149338439137309795, 8.706765597420159753229260194052, 9.538291305055480850556186888532, 9.895562633630313165325886248135, 10.82259124853772396204677861897, 11.32775537258475749001289496612, 11.65711565681209698953976937280, 12.57734678591887601772847316331, 12.86712142261172265095003988916
