| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 9-s − 4·14-s + 16-s + 17-s + 18-s + 11·23-s − 2·25-s − 4·28-s + 9·31-s + 32-s + 34-s + 36-s − 10·41-s + 11·46-s + 12·47-s + 9·49-s − 2·50-s − 4·56-s + 9·62-s − 4·63-s + 64-s + 68-s + 17·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 2.29·23-s − 2/5·25-s − 0.755·28-s + 1.61·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 1.56·41-s + 1.62·46-s + 1.75·47-s + 9/7·49-s − 0.282·50-s − 0.534·56-s + 1.14·62-s − 0.503·63-s + 1/8·64-s + 0.121·68-s + 2.01·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.143310911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.143310911\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−9.718219353792715358038687607992, −8.952325910520837235626941157947, −8.724605520310189486607045679673, −7.960914621459554270832019976707, −7.30513121604757218594510797594, −6.85596906221345784659348396929, −6.60684869400441825562165673213, −5.95491904478088902584726881234, −5.40746075153588595542019183329, −4.81645400250173160000113568885, −4.19416202378143746500150885957, −3.43741424618277769747505800724, −3.06125593089125241382315475542, −2.34407269968672403867201831717, −0.997725468994788639225681247326,
0.997725468994788639225681247326, 2.34407269968672403867201831717, 3.06125593089125241382315475542, 3.43741424618277769747505800724, 4.19416202378143746500150885957, 4.81645400250173160000113568885, 5.40746075153588595542019183329, 5.95491904478088902584726881234, 6.60684869400441825562165673213, 6.85596906221345784659348396929, 7.30513121604757218594510797594, 7.960914621459554270832019976707, 8.724605520310189486607045679673, 8.952325910520837235626941157947, 9.718219353792715358038687607992
