| L(s) = 1 | + 4-s − 2·9-s + 2·29-s − 2·36-s − 6·41-s + 2·49-s − 2·61-s − 64-s + 81-s + 2·101-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 4-s − 2·9-s + 2·29-s − 2·36-s − 6·41-s + 2·49-s − 2·61-s − 64-s + 81-s + 2·101-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 6·164-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8148643648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8148643648\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−7.14037635481571728803830361910, −6.95227047884737193029526882317, −6.61906794277114016186520961428, −6.32674885614868403946017032369, −6.17630745858270652205558325178, −6.14890934746873533668125538169, −6.06045764698935826266215100054, −5.39354112077187983942667014025, −5.28037990105951589422231124115, −5.24958381374050162845032207047, −4.97789700812871558472027256308, −4.76677061721118435466180975508, −4.34539019164284771742773514157, −4.23232193270706904981025859990, −3.79941302656713505683039831712, −3.39304892487479526626106812516, −3.36770677875284724662857555813, −3.08122157290271894025868250804, −2.78156080178599619095024879941, −2.67701923943017032441066246654, −2.32868765934268631717401736714, −1.83738828195789329246086905841, −1.72276230284197702806121687226, −1.37615569265469165841756624238, −0.55856208997797973395082245888,
0.55856208997797973395082245888, 1.37615569265469165841756624238, 1.72276230284197702806121687226, 1.83738828195789329246086905841, 2.32868765934268631717401736714, 2.67701923943017032441066246654, 2.78156080178599619095024879941, 3.08122157290271894025868250804, 3.36770677875284724662857555813, 3.39304892487479526626106812516, 3.79941302656713505683039831712, 4.23232193270706904981025859990, 4.34539019164284771742773514157, 4.76677061721118435466180975508, 4.97789700812871558472027256308, 5.24958381374050162845032207047, 5.28037990105951589422231124115, 5.39354112077187983942667014025, 6.06045764698935826266215100054, 6.14890934746873533668125538169, 6.17630745858270652205558325178, 6.32674885614868403946017032369, 6.61906794277114016186520961428, 6.95227047884737193029526882317, 7.14037635481571728803830361910
