L(s) = 1 | + 4-s + 7-s − 25-s + 28-s − 4·37-s + 2·43-s − 64-s − 2·67-s − 2·79-s − 100-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯ |
L(s) = 1 | + 4-s + 7-s − 25-s + 28-s − 4·37-s + 2·43-s − 64-s − 2·67-s − 2·79-s − 100-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 2·172-s + 173-s − 175-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.057627678\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.057627678\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.99940424432874667142901200152, −10.90247207123009149946381141655, −10.37637298825697509188547777643, −10.11069840414037343952479266800, −9.215880219020450968942326803420, −9.128346622071799831094325365762, −8.334820930453629618858090074600, −8.222153128999321471228880011665, −7.40060848009186530620805562733, −7.25944728115866275533600091715, −6.80111424525177415749482258036, −6.20764933553850848495352885464, −5.50263309187791770556900334756, −5.44727404574104740297571047752, −4.50423450333269919520417536148, −4.19673234047705220604607422569, −3.34247075571284060273309278450, −2.80538349100618224694946826426, −1.86503485769790619053707558087, −1.67920132820486876628609592671,
1.67920132820486876628609592671, 1.86503485769790619053707558087, 2.80538349100618224694946826426, 3.34247075571284060273309278450, 4.19673234047705220604607422569, 4.50423450333269919520417536148, 5.44727404574104740297571047752, 5.50263309187791770556900334756, 6.20764933553850848495352885464, 6.80111424525177415749482258036, 7.25944728115866275533600091715, 7.40060848009186530620805562733, 8.222153128999321471228880011665, 8.334820930453629618858090074600, 9.128346622071799831094325365762, 9.215880219020450968942326803420, 10.11069840414037343952479266800, 10.37637298825697509188547777643, 10.90247207123009149946381141655, 10.99940424432874667142901200152