| L(s) = 1 | − 3-s − 3·4-s − 8·7-s + 9-s + 3·12-s + 4·13-s + 5·16-s + 8·21-s − 6·25-s − 27-s + 24·28-s − 16·31-s − 3·36-s + 12·37-s − 4·39-s − 5·48-s + 34·49-s − 12·52-s − 12·61-s − 8·63-s − 3·64-s − 8·67-s + 28·73-s + 6·75-s + 8·79-s + 81-s − 24·84-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 3.02·7-s + 1/3·9-s + 0.866·12-s + 1.10·13-s + 5/4·16-s + 1.74·21-s − 6/5·25-s − 0.192·27-s + 4.53·28-s − 2.87·31-s − 1/2·36-s + 1.97·37-s − 0.640·39-s − 0.721·48-s + 34/7·49-s − 1.66·52-s − 1.53·61-s − 1.00·63-s − 3/8·64-s − 0.977·67-s + 3.27·73-s + 0.692·75-s + 0.900·79-s + 1/9·81-s − 2.61·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395307 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395307 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.829662012954888204160857557729, −7.77744707656776841753907290662, −7.67376126887882242787787369865, −6.85553577597705614902881023761, −6.38866021531112361683475479825, −6.11982556539470572208453322290, −5.66464594246847012421851082372, −5.25909302550772251578024339366, −4.33065348032451946577135661956, −3.98994534926773298366607994601, −3.36662241945177810035682717763, −3.31119773648628104035349562784, −2.09666333780189433867543562159, −0.71891279628917143719551815370, 0,
0.71891279628917143719551815370, 2.09666333780189433867543562159, 3.31119773648628104035349562784, 3.36662241945177810035682717763, 3.98994534926773298366607994601, 4.33065348032451946577135661956, 5.25909302550772251578024339366, 5.66464594246847012421851082372, 6.11982556539470572208453322290, 6.38866021531112361683475479825, 6.85553577597705614902881023761, 7.67376126887882242787787369865, 7.77744707656776841753907290662, 8.829662012954888204160857557729
