| L(s) = 1 | − 3-s − 5-s − 7-s − 2·11-s + 13-s + 15-s − 2·17-s + 21-s + 27-s + 29-s + 2·33-s + 35-s − 39-s − 2·47-s + 2·51-s + 2·55-s − 65-s − 2·71-s − 2·73-s + 2·77-s − 2·79-s − 81-s + 83-s + 2·85-s − 87-s − 91-s − 2·97-s + ⋯ |
| L(s) = 1 | − 3-s − 5-s − 7-s − 2·11-s + 13-s + 15-s − 2·17-s + 21-s + 27-s + 29-s + 2·33-s + 35-s − 39-s − 2·47-s + 2·51-s + 2·55-s − 65-s − 2·71-s − 2·73-s + 2·77-s − 2·79-s − 81-s + 83-s + 2·85-s − 87-s − 91-s − 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1708496611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1708496611\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.17571133057321982192217242226, −9.960404819097634586826001883482, −9.161810921855768919372592667284, −8.813447934972045669805243853457, −8.369207584681169553210032516886, −8.209836644865922031817893005036, −7.60352544086828354206564663855, −7.12834963442663261870102134463, −6.78640693361545804290467268543, −6.29577245621593577565232758370, −5.97098949285781605426479310695, −5.58986539697941753211106472356, −4.95586912477766349241103116159, −4.45849541859484611406644373857, −4.35778895576497876849218120891, −3.38628760539103877969484622366, −3.05916145408168353942793336503, −2.60576088608570906903360334815, −1.71354733364478025856655651242, −0.36539991274136573843907421278,
0.36539991274136573843907421278, 1.71354733364478025856655651242, 2.60576088608570906903360334815, 3.05916145408168353942793336503, 3.38628760539103877969484622366, 4.35778895576497876849218120891, 4.45849541859484611406644373857, 4.95586912477766349241103116159, 5.58986539697941753211106472356, 5.97098949285781605426479310695, 6.29577245621593577565232758370, 6.78640693361545804290467268543, 7.12834963442663261870102134463, 7.60352544086828354206564663855, 8.209836644865922031817893005036, 8.369207584681169553210032516886, 8.813447934972045669805243853457, 9.161810921855768919372592667284, 9.960404819097634586826001883482, 10.17571133057321982192217242226
