| L(s) = 1 | − 2-s + 8-s − 16-s + 2·17-s − 2·34-s + 2·49-s + 2·53-s + 2·61-s + 64-s − 2·98-s − 2·106-s + 2·109-s + 4·113-s + 2·121-s − 2·122-s + 127-s − 128-s + 131-s + 2·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + ⋯ |
| L(s) = 1 | − 2-s + 8-s − 16-s + 2·17-s − 2·34-s + 2·49-s + 2·53-s + 2·61-s + 64-s − 2·98-s − 2·106-s + 2·109-s + 4·113-s + 2·121-s − 2·122-s + 127-s − 128-s + 131-s + 2·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8394737099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8394737099\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.073899166379738248810567422432, −8.872199744927892487379817206518, −8.415922824338715098216996494901, −8.269865490636535312864902379023, −7.67543391588187900490736525016, −7.43609004240879064963050263962, −7.07813709442973736853300817084, −6.85719080956249332986913979381, −5.96975410646137870802310270271, −5.81855758999976185633811029531, −5.45233726070309140685976454346, −4.89086501950668731663202192182, −4.57330612341103889223627231866, −3.94378784983662353489005582332, −3.58796985750289119547458956224, −3.21791612930898938854884587085, −2.26921688891191251507911889402, −2.18783905301853992838263396990, −1.09760211608804401461650525018, −0.906711762958472861950307308783,
0.906711762958472861950307308783, 1.09760211608804401461650525018, 2.18783905301853992838263396990, 2.26921688891191251507911889402, 3.21791612930898938854884587085, 3.58796985750289119547458956224, 3.94378784983662353489005582332, 4.57330612341103889223627231866, 4.89086501950668731663202192182, 5.45233726070309140685976454346, 5.81855758999976185633811029531, 5.96975410646137870802310270271, 6.85719080956249332986913979381, 7.07813709442973736853300817084, 7.43609004240879064963050263962, 7.67543391588187900490736525016, 8.269865490636535312864902379023, 8.415922824338715098216996494901, 8.872199744927892487379817206518, 9.073899166379738248810567422432
