| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s − 2·9-s + 3·14-s + 16-s − 3·17-s − 2·18-s + 5·23-s + 3·25-s + 3·28-s + 31-s + 32-s − 3·34-s − 2·36-s + 10·41-s + 5·46-s + 6·47-s + 5·49-s + 3·50-s + 3·56-s + 62-s − 6·63-s + 64-s − 3·68-s + 7·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.04·23-s + 3/5·25-s + 0.566·28-s + 0.179·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 1.56·41-s + 0.737·46-s + 0.875·47-s + 5/7·49-s + 0.424·50-s + 0.400·56-s + 0.127·62-s − 0.755·63-s + 1/8·64-s − 0.363·68-s + 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.840416380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.840416380\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 145 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.203066449939682583633459123551, −8.659553993511094722343839334835, −8.382428443249622923386567132691, −7.74574540887451372716773499309, −7.27984934428373185671500794005, −6.83444557479384005948671261698, −6.19420325908539149642828294216, −5.63827305286270959936232483056, −5.24643573781373602512633228945, −4.52960211247326271349417418098, −4.34156500772589221190936313032, −3.41414189914333366430531928542, −2.72633232227990318943454424086, −2.15319592426029789347360995595, −1.09734218504033192800030640053,
1.09734218504033192800030640053, 2.15319592426029789347360995595, 2.72633232227990318943454424086, 3.41414189914333366430531928542, 4.34156500772589221190936313032, 4.52960211247326271349417418098, 5.24643573781373602512633228945, 5.63827305286270959936232483056, 6.19420325908539149642828294216, 6.83444557479384005948671261698, 7.27984934428373185671500794005, 7.74574540887451372716773499309, 8.382428443249622923386567132691, 8.659553993511094722343839334835, 9.203066449939682583633459123551
