| L(s) = 1 | − 2·5-s − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·5-s − 9-s − 2·17-s + 3·25-s + 2·37-s + 2·45-s − 2·49-s + 2·53-s + 2·61-s + 2·73-s + 81-s + 4·85-s + 2·89-s − 2·101-s − 2·109-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1812040511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1812040511\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−9.309996863911135671874973673591, −8.897422967595805715172683466262, −8.499984918113276897785770765506, −8.294787861306813368008114044733, −8.215630286176052104965475202038, −8.152010468097277936804154069629, −7.64951924974791220569551434192, −7.30376574637874454210651227516, −7.10972948766242002299187214914, −6.96509250196731627601181733578, −6.37343180237093145708937355264, −6.27309012268999072264489363596, −6.25005304233019844508058994269, −5.57705880420581847333263358842, −5.04202855766530028312325826175, −4.97377363559341987547388628072, −4.86158159448459267204310288019, −4.23739228702478708277672850209, −3.85082022785688915825160501435, −3.84728006389649320724127329225, −3.51637984074493596937248463765, −2.76788953705767214556321314744, −2.50306402926853600546352933742, −2.37913706468566554218222193660, −1.18115060432250576025081445514,
1.18115060432250576025081445514, 2.37913706468566554218222193660, 2.50306402926853600546352933742, 2.76788953705767214556321314744, 3.51637984074493596937248463765, 3.84728006389649320724127329225, 3.85082022785688915825160501435, 4.23739228702478708277672850209, 4.86158159448459267204310288019, 4.97377363559341987547388628072, 5.04202855766530028312325826175, 5.57705880420581847333263358842, 6.25005304233019844508058994269, 6.27309012268999072264489363596, 6.37343180237093145708937355264, 6.96509250196731627601181733578, 7.10972948766242002299187214914, 7.30376574637874454210651227516, 7.64951924974791220569551434192, 8.152010468097277936804154069629, 8.215630286176052104965475202038, 8.294787861306813368008114044733, 8.499984918113276897785770765506, 8.897422967595805715172683466262, 9.309996863911135671874973673591
