| L(s) = 1 | − 2-s + 4-s + 5-s − 2·8-s − 10-s + 2·16-s + 2·17-s − 2·19-s + 20-s − 23-s + 31-s − 2·32-s − 2·34-s + 2·38-s − 2·40-s + 46-s + 2·47-s − 49-s + 2·53-s + 61-s − 62-s + 3·64-s + 2·68-s − 2·76-s + 79-s + 2·80-s − 83-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 5-s − 2·8-s − 10-s + 2·16-s + 2·17-s − 2·19-s + 20-s − 23-s + 31-s − 2·32-s − 2·34-s + 2·38-s − 2·40-s + 46-s + 2·47-s − 49-s + 2·53-s + 61-s − 62-s + 3·64-s + 2·68-s − 2·76-s + 79-s + 2·80-s − 83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4956418988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4956418988\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−11.75592060411583780117775129241, −11.12919523441986864015023418405, −10.58023505175654308698770872629, −10.17139498500539334332012494454, −9.926452438481325881254636165547, −9.573162261905505329327860118547, −8.803308771221547068513593430479, −8.761493075740315147376689294646, −8.019681034380590845444153519589, −7.79060459310188126284096061338, −6.99376370935908539309254717851, −6.38617212887298080680866494517, −6.23157570783589567837452934809, −5.50020971905860274164625045316, −5.40367825166858019471337999689, −4.09584441689670257259431011088, −3.64897770546299677739154530805, −2.56436313690679554797468672672, −2.41658163365104649263169340771, −1.30758115796568570267697600701,
1.30758115796568570267697600701, 2.41658163365104649263169340771, 2.56436313690679554797468672672, 3.64897770546299677739154530805, 4.09584441689670257259431011088, 5.40367825166858019471337999689, 5.50020971905860274164625045316, 6.23157570783589567837452934809, 6.38617212887298080680866494517, 6.99376370935908539309254717851, 7.79060459310188126284096061338, 8.019681034380590845444153519589, 8.761493075740315147376689294646, 8.803308771221547068513593430479, 9.573162261905505329327860118547, 9.926452438481325881254636165547, 10.17139498500539334332012494454, 10.58023505175654308698770872629, 11.12919523441986864015023418405, 11.75592060411583780117775129241
