L(s) = 1 | − 2·2-s + 3·4-s + 2·7-s − 4·8-s − 2·9-s − 4·13-s − 4·14-s + 5·16-s + 4·18-s − 5·25-s + 8·26-s + 6·28-s − 12·29-s − 6·32-s − 6·36-s + 4·37-s − 24·47-s + 3·49-s + 10·50-s − 12·52-s − 8·56-s + 24·58-s + 16·61-s − 4·63-s + 7·64-s − 8·67-s + 8·72-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s − 2/3·9-s − 1.10·13-s − 1.06·14-s + 5/4·16-s + 0.942·18-s − 25-s + 1.56·26-s + 1.13·28-s − 2.22·29-s − 1.06·32-s − 36-s + 0.657·37-s − 3.50·47-s + 3/7·49-s + 1.41·50-s − 1.66·52-s − 1.06·56-s + 3.15·58-s + 2.04·61-s − 0.503·63-s + 7/8·64-s − 0.977·67-s + 0.942·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4869251355\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4869251355\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.149115589354817244986775905302, −7.921942048273572677150608948373, −7.57571100088867902110310233811, −6.96023036898751402073285166387, −6.75497369172733229312582987288, −6.00728441107086114890911949727, −5.57928681742950427486583645839, −5.23472501559197949370067568971, −4.58610841451955891029930986059, −3.91813300576011100108011679413, −3.27110855367138212573533050585, −2.67579223441987319029898681629, −1.94147475247196906907387077179, −1.67889512896363126345389610058, −0.40214508136838586607328490895,
0.40214508136838586607328490895, 1.67889512896363126345389610058, 1.94147475247196906907387077179, 2.67579223441987319029898681629, 3.27110855367138212573533050585, 3.91813300576011100108011679413, 4.58610841451955891029930986059, 5.23472501559197949370067568971, 5.57928681742950427486583645839, 6.00728441107086114890911949727, 6.75497369172733229312582987288, 6.96023036898751402073285166387, 7.57571100088867902110310233811, 7.921942048273572677150608948373, 8.149115589354817244986775905302