| L(s) = 1 | + 2-s − 4-s − 4·5-s − 3·8-s + 3·9-s − 4·10-s − 2·13-s − 16-s + 4·17-s + 3·18-s + 4·20-s + 3·25-s − 2·26-s − 2·29-s + 5·32-s + 4·34-s − 3·36-s + 4·37-s + 12·40-s − 12·41-s − 12·45-s − 49-s + 3·50-s + 2·52-s + 10·53-s − 2·58-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s − 1.06·8-s + 9-s − 1.26·10-s − 0.554·13-s − 1/4·16-s + 0.970·17-s + 0.707·18-s + 0.894·20-s + 3/5·25-s − 0.392·26-s − 0.371·29-s + 0.883·32-s + 0.685·34-s − 1/2·36-s + 0.657·37-s + 1.89·40-s − 1.87·41-s − 1.78·45-s − 1/7·49-s + 0.424·50-s + 0.277·52-s + 1.37·53-s − 0.262·58-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 41 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 27 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 97 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−7.58991402882434767176800950547, −7.35392594568385346862560569400, −7.16996060519236524570317010242, −6.25732656422371547403268186266, −6.12117328897382878839267077346, −5.21302877583932413947222007031, −5.08281477433864113140272604013, −4.52199048449353000563317420658, −3.99893912434989633280304918243, −3.79469965445300721118601646658, −3.36423607000611598911854034589, −2.75305470475718881800757537055, −1.86633571095393736222123431096, −0.881482214760266314804723977236, 0,
0.881482214760266314804723977236, 1.86633571095393736222123431096, 2.75305470475718881800757537055, 3.36423607000611598911854034589, 3.79469965445300721118601646658, 3.99893912434989633280304918243, 4.52199048449353000563317420658, 5.08281477433864113140272604013, 5.21302877583932413947222007031, 6.12117328897382878839267077346, 6.25732656422371547403268186266, 7.16996060519236524570317010242, 7.35392594568385346862560569400, 7.58991402882434767176800950547
