L(s) = 1 | − 2·7-s + 2·13-s + 25-s + 3·31-s + 37-s + 3·49-s + 61-s + 3·67-s + 2·73-s − 3·79-s − 4·91-s + 2·97-s + 3·103-s − 109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s − 2·175-s + ⋯ |
L(s) = 1 | − 2·7-s + 2·13-s + 25-s + 3·31-s + 37-s + 3·49-s + 61-s + 3·67-s + 2·73-s − 3·79-s − 4·91-s + 2·97-s + 3·103-s − 109-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s − 2·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.356202399\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.356202399\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−8.964051542748206565594137672453, −8.694900962424661122857533648938, −8.422613933294693839428940217796, −8.195841552536736986962084189561, −7.41380445650433216744209111051, −7.26984110398530456153931427144, −6.54354476762319116323897759549, −6.45851359507801734099783236945, −6.10733307279266288330178701927, −6.05580078074835682221721177763, −5.10035387925114202314747347158, −5.06084949260165392597210493948, −4.09290699334847601988498339620, −4.08080041811156436383945455630, −3.34652332016447491344779235242, −3.30435502885153488796315971283, −2.45354232716552858738051869983, −2.44425578812246109941740019186, −1.02213249716943992987137170446, −0.982103471695430954523385122772,
0.982103471695430954523385122772, 1.02213249716943992987137170446, 2.44425578812246109941740019186, 2.45354232716552858738051869983, 3.30435502885153488796315971283, 3.34652332016447491344779235242, 4.08080041811156436383945455630, 4.09290699334847601988498339620, 5.06084949260165392597210493948, 5.10035387925114202314747347158, 6.05580078074835682221721177763, 6.10733307279266288330178701927, 6.45851359507801734099783236945, 6.54354476762319116323897759549, 7.26984110398530456153931427144, 7.41380445650433216744209111051, 8.195841552536736986962084189561, 8.422613933294693839428940217796, 8.694900962424661122857533648938, 8.964051542748206565594137672453