| L(s) = 1 | − 9-s + 4·13-s − 2·17-s − 4·25-s + 2·29-s − 2·37-s − 2·41-s − 49-s + 2·61-s + 81-s + 2·89-s + 2·97-s − 2·101-s − 2·113-s − 4·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | − 9-s + 4·13-s − 2·17-s − 4·25-s + 2·29-s − 2·37-s − 2·41-s − 49-s + 2·61-s + 81-s + 2·89-s + 2·97-s − 2·101-s − 2·113-s − 4·117-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4725620121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4725620121\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.362350221878164128253945514060, −8.157227439348933907829570435057, −8.082139972157866462866888207237, −7.77131598109035245563120524235, −7.52366621259775273015933246989, −6.78913577969456354190091977661, −6.66474908410843033072042329246, −6.66415214941330116165408095173, −6.53658872710570387340612681246, −6.01198635240913829305323751910, −5.96317286444364306479759053007, −5.55519337493800148989496338698, −5.50067776687963736586586474262, −5.12924886940871015484160370661, −4.67881471752009105687629546664, −4.31431976680478807299938074315, −4.09853388883791422399930173828, −3.80493148432603117085106764238, −3.53034927906849868206101622665, −3.29670150837080691439743886181, −3.09689828821238581465743595102, −2.28266084185552703061708015507, −2.01481993602685396123698186653, −1.79150581374722169523346327457, −1.16817482024152375815029543823,
1.16817482024152375815029543823, 1.79150581374722169523346327457, 2.01481993602685396123698186653, 2.28266084185552703061708015507, 3.09689828821238581465743595102, 3.29670150837080691439743886181, 3.53034927906849868206101622665, 3.80493148432603117085106764238, 4.09853388883791422399930173828, 4.31431976680478807299938074315, 4.67881471752009105687629546664, 5.12924886940871015484160370661, 5.50067776687963736586586474262, 5.55519337493800148989496338698, 5.96317286444364306479759053007, 6.01198635240913829305323751910, 6.53658872710570387340612681246, 6.66415214941330116165408095173, 6.66474908410843033072042329246, 6.78913577969456354190091977661, 7.52366621259775273015933246989, 7.77131598109035245563120524235, 8.082139972157866462866888207237, 8.157227439348933907829570435057, 8.362350221878164128253945514060
