L(s) = 1 | − 4·3-s + 4-s + 2·7-s + 6·9-s − 4·12-s + 16-s − 8·21-s − 10·25-s + 4·27-s + 2·28-s + 6·36-s + 2·37-s + 12·41-s − 24·47-s − 4·48-s + 3·49-s + 12·53-s + 12·63-s + 64-s − 8·67-s + 4·73-s + 40·75-s − 37·81-s − 12·83-s − 8·84-s − 10·100-s + 24·107-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s + 0.755·7-s + 2·9-s − 1.15·12-s + 1/4·16-s − 1.74·21-s − 2·25-s + 0.769·27-s + 0.377·28-s + 36-s + 0.328·37-s + 1.87·41-s − 3.50·47-s − 0.577·48-s + 3/7·49-s + 1.64·53-s + 1.51·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 4.61·75-s − 4.11·81-s − 1.31·83-s − 0.872·84-s − 100-s + 2.32·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 268324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 268324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.485152769343560672100851205130, −8.169998867413597699811066785828, −7.57571100088867902110310233811, −7.16851316689893371261782075681, −6.45499171287041897802749746589, −6.26080129812508362760024888239, −5.61058899029416894793643414289, −5.57928681742950427486583645839, −4.81446031852430885616934703988, −4.47617404964643867507446377668, −3.72847359898829464933007778606, −2.82267866440533335693556734961, −1.97153093819951005096748677540, −1.10321404612740190875240729674, 0,
1.10321404612740190875240729674, 1.97153093819951005096748677540, 2.82267866440533335693556734961, 3.72847359898829464933007778606, 4.47617404964643867507446377668, 4.81446031852430885616934703988, 5.57928681742950427486583645839, 5.61058899029416894793643414289, 6.26080129812508362760024888239, 6.45499171287041897802749746589, 7.16851316689893371261782075681, 7.57571100088867902110310233811, 8.169998867413597699811066785828, 8.485152769343560672100851205130