L(s) = 1 | + 4-s − 9-s − 4·11-s + 2·19-s − 36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s + 4·89-s + 4·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 2·171-s + ⋯ |
L(s) = 1 | + 4-s − 9-s − 4·11-s + 2·19-s − 36-s + 2·41-s − 4·44-s − 4·49-s − 2·59-s − 64-s + 2·76-s + 81-s + 4·89-s + 4·99-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 2·169-s − 2·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5240511406\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5240511406\) |
\(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.17239023837203753593964451194, −6.00875899359188988154341987657, −5.79830443028639570780183423752, −5.67209296982242134807563279161, −5.26571185475875722080018992640, −5.09369246435461083925128998982, −5.08583679416755380118742700208, −4.96694254318839599471162262452, −4.90998552981309850532456330906, −4.41663682811393252382404712365, −4.14042401226337320820489472849, −4.12743118762662828247110578370, −3.46929155713256913925837996333, −3.42188949055513843544158764582, −3.01080050816064212321032002398, −2.97607261905949125944161295219, −2.97274488362386089879285307783, −2.82838289726690583673225602858, −2.26224199376516917850708167908, −2.21192457579688578550143415161, −1.92268295060124838718515251018, −1.83794377455359326389843473585, −1.13594859349304017073477690312, −0.979565346964776444200058671677, −0.26214172676901085807433587199,
0.26214172676901085807433587199, 0.979565346964776444200058671677, 1.13594859349304017073477690312, 1.83794377455359326389843473585, 1.92268295060124838718515251018, 2.21192457579688578550143415161, 2.26224199376516917850708167908, 2.82838289726690583673225602858, 2.97274488362386089879285307783, 2.97607261905949125944161295219, 3.01080050816064212321032002398, 3.42188949055513843544158764582, 3.46929155713256913925837996333, 4.12743118762662828247110578370, 4.14042401226337320820489472849, 4.41663682811393252382404712365, 4.90998552981309850532456330906, 4.96694254318839599471162262452, 5.08583679416755380118742700208, 5.09369246435461083925128998982, 5.26571185475875722080018992640, 5.67209296982242134807563279161, 5.79830443028639570780183423752, 6.00875899359188988154341987657, 6.17239023837203753593964451194