| L(s) = 1 | + 2-s − 9-s + 2·13-s + 2·17-s − 18-s + 2·26-s − 2·29-s − 32-s + 2·34-s − 3·37-s − 2·41-s + 4·49-s − 3·53-s − 2·58-s − 2·61-s − 64-s + 2·73-s − 3·74-s − 2·82-s + 3·89-s + 2·97-s + 4·98-s − 2·101-s − 3·106-s − 2·109-s − 3·113-s − 2·117-s + ⋯ |
| L(s) = 1 | + 2-s − 9-s + 2·13-s + 2·17-s − 18-s + 2·26-s − 2·29-s − 32-s + 2·34-s − 3·37-s − 2·41-s + 4·49-s − 3·53-s − 2·58-s − 2·61-s − 64-s + 2·73-s − 3·74-s − 2·82-s + 3·89-s + 2·97-s + 4·98-s − 2·101-s − 3·106-s − 2·109-s − 3·113-s − 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8616765881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8616765881\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
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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
−8.325218650228243729722808444338, −7.66781577251715711708457081064, −7.62933430497668731439454452980, −7.62044951110699001140253523361, −7.07292977975925877799934291376, −6.95358196106255122562595302646, −6.61071024389396147058275865079, −6.25660396517568648205399644964, −6.08020554531281454284609123230, −5.88680666185041904942053912371, −5.58644150523634675874638064398, −5.38674994216360083097844497251, −5.10011139322546652559662997938, −4.90127302739369193307540039603, −4.88940007036772082646059296898, −3.98718938395307868810508543941, −3.92322853445960267329599222447, −3.78052381227207248570949005381, −3.54382219627713142150204716001, −3.22864378461474225646871697236, −2.99970536277783307009686114347, −2.50481158892233231583573835780, −1.83690840557181911574673808019, −1.68175807421680079578803111696, −1.22639904476965981640667428583,
1.22639904476965981640667428583, 1.68175807421680079578803111696, 1.83690840557181911574673808019, 2.50481158892233231583573835780, 2.99970536277783307009686114347, 3.22864378461474225646871697236, 3.54382219627713142150204716001, 3.78052381227207248570949005381, 3.92322853445960267329599222447, 3.98718938395307868810508543941, 4.88940007036772082646059296898, 4.90127302739369193307540039603, 5.10011139322546652559662997938, 5.38674994216360083097844497251, 5.58644150523634675874638064398, 5.88680666185041904942053912371, 6.08020554531281454284609123230, 6.25660396517568648205399644964, 6.61071024389396147058275865079, 6.95358196106255122562595302646, 7.07292977975925877799934291376, 7.62044951110699001140253523361, 7.62933430497668731439454452980, 7.66781577251715711708457081064, 8.325218650228243729722808444338
