| L(s) = 1 | + 2-s + 4-s + 7-s − 8-s + 4·9-s + 7·11-s + 14-s − 3·16-s + 4·18-s + 7·22-s − 6·23-s + 25-s + 28-s + 8·29-s − 5·32-s + 4·36-s − 7·37-s + 7·43-s + 7·44-s − 6·46-s − 6·49-s + 50-s − 5·53-s − 56-s + 8·58-s + 4·63-s − 3·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 4/3·9-s + 2.11·11-s + 0.267·14-s − 3/4·16-s + 0.942·18-s + 1.49·22-s − 1.25·23-s + 1/5·25-s + 0.188·28-s + 1.48·29-s − 0.883·32-s + 2/3·36-s − 1.15·37-s + 1.06·43-s + 1.05·44-s − 0.884·46-s − 6/7·49-s + 0.141·50-s − 0.686·53-s − 0.133·56-s + 1.05·58-s + 0.503·63-s − 3/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118874 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118874 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.970744811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.970744811\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 - p T + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 1213 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 105 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
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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
−9.346784135207533132098226225759, −9.154945899057777724749949488247, −8.432800128472359896605771344071, −8.006024004147813282044244627427, −7.21340306553568719306764419128, −6.86372714276601150409307329103, −6.44002784757107748742582825719, −6.04212991834912406516080478903, −5.30980329239655386259630904106, −4.47790908732243119328415477396, −4.28681925656410329831867685566, −3.70426372772003518686689566835, −2.97857373972891683233201248727, −1.92319725559422056022061253720, −1.30682626814850371691193290694,
1.30682626814850371691193290694, 1.92319725559422056022061253720, 2.97857373972891683233201248727, 3.70426372772003518686689566835, 4.28681925656410329831867685566, 4.47790908732243119328415477396, 5.30980329239655386259630904106, 6.04212991834912406516080478903, 6.44002784757107748742582825719, 6.86372714276601150409307329103, 7.21340306553568719306764419128, 8.006024004147813282044244627427, 8.432800128472359896605771344071, 9.154945899057777724749949488247, 9.346784135207533132098226225759
