L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 6·13-s + 16-s − 6·25-s − 27-s + 36-s − 6·39-s + 8·43-s − 48-s + 10·49-s + 6·52-s − 4·61-s + 64-s + 6·75-s + 81-s − 6·100-s − 32·103-s − 108-s + 6·117-s − 22·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/2·4-s + 1/3·9-s − 0.288·12-s + 1.66·13-s + 1/4·16-s − 6/5·25-s − 0.192·27-s + 1/6·36-s − 0.960·39-s + 1.21·43-s − 0.144·48-s + 10/7·49-s + 0.832·52-s − 0.512·61-s + 1/8·64-s + 0.692·75-s + 1/9·81-s − 3/5·100-s − 3.15·103-s − 0.0962·108-s + 0.554·117-s − 2·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18252 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.112282735\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.112282735\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−10.92591047448404489455298703812, −10.57457186614303466943177704363, −10.06454957650976622915451112660, −9.253220401756145526066424253394, −8.893772573450384569097262787010, −8.041746957490896078081456984786, −7.68916893113167905967774177448, −6.87944639643402688270762012935, −6.34006998241373302303396063951, −5.81225327835322207959592349923, −5.31396272181510219841318127340, −4.17395035154105580618519439241, −3.74904387488260970946838288229, −2.59439103665810412476981586962, −1.37286813188261880910767163756,
1.37286813188261880910767163756, 2.59439103665810412476981586962, 3.74904387488260970946838288229, 4.17395035154105580618519439241, 5.31396272181510219841318127340, 5.81225327835322207959592349923, 6.34006998241373302303396063951, 6.87944639643402688270762012935, 7.68916893113167905967774177448, 8.041746957490896078081456984786, 8.893772573450384569097262787010, 9.253220401756145526066424253394, 10.06454957650976622915451112660, 10.57457186614303466943177704363, 10.92591047448404489455298703812