| L(s) = 1 | − 3-s + 9-s − 8·11-s − 12·13-s − 8·23-s − 10·25-s − 27-s + 8·33-s − 4·37-s + 12·39-s + 24·47-s − 10·49-s − 8·59-s − 4·61-s + 8·69-s + 8·71-s − 20·73-s + 10·75-s + 81-s + 24·83-s − 12·97-s − 8·99-s − 24·107-s + 4·109-s + 4·111-s − 12·117-s + 26·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 2.41·11-s − 3.32·13-s − 1.66·23-s − 2·25-s − 0.192·27-s + 1.39·33-s − 0.657·37-s + 1.92·39-s + 3.50·47-s − 1.42·49-s − 1.04·59-s − 0.512·61-s + 0.963·69-s + 0.949·71-s − 2.34·73-s + 1.15·75-s + 1/9·81-s + 2.63·83-s − 1.21·97-s − 0.804·99-s − 2.32·107-s + 0.383·109-s + 0.379·111-s − 1.10·117-s + 2.36·121-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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.86636283231227873209414758068, −7.73185283482156320096864018539, −7.29744951786402483785793531451, −7.00186001179257888349709337610, −5.96427857852485389909948133454, −5.81271121311664380914911812156, −5.25073137830748218691985779847, −4.86024035915155668889978626292, −4.46736447254899856037450817253, −3.77490371552045331982722842815, −2.79778517840079752431830788770, −2.36547685273488160999702672858, −2.02071693225675944817521065685, 0, 0,
2.02071693225675944817521065685, 2.36547685273488160999702672858, 2.79778517840079752431830788770, 3.77490371552045331982722842815, 4.46736447254899856037450817253, 4.86024035915155668889978626292, 5.25073137830748218691985779847, 5.81271121311664380914911812156, 5.96427857852485389909948133454, 7.00186001179257888349709337610, 7.29744951786402483785793531451, 7.73185283482156320096864018539, 7.86636283231227873209414758068
