L(s) = 1 | − 4·25-s + 2·49-s − 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | − 4·25-s + 2·49-s − 4·73-s + 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7296794513\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7296794513\) |
\(L(1)\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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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
−6.99326595092749575480987268386, −6.45943846897056164066551919052, −6.36512830745111480021899452796, −6.10369930287206406779904534040, −6.00036035500522940460046212826, −5.90508806644719562779067785606, −5.51353532999741307869215834485, −5.45725681129531527842857408575, −5.26162929618539286342752890034, −4.69007782614118770680168609932, −4.65168210636875656123279438289, −4.59546122557385217629789820771, −4.08379331213660696936032823688, −3.87590463956139070015727376229, −3.80692219246757224978804995502, −3.63427805708805487319670145154, −3.25591163728466483437690393791, −2.99981303877586771403430792425, −2.48014192037986862179197496554, −2.38737495484767966059098893467, −2.31466449259314218333480453223, −1.72280138499637881989110314377, −1.53209581930402943391581363298, −1.27374928404865229311054650625, −0.46338051862625487952607940924,
0.46338051862625487952607940924, 1.27374928404865229311054650625, 1.53209581930402943391581363298, 1.72280138499637881989110314377, 2.31466449259314218333480453223, 2.38737495484767966059098893467, 2.48014192037986862179197496554, 2.99981303877586771403430792425, 3.25591163728466483437690393791, 3.63427805708805487319670145154, 3.80692219246757224978804995502, 3.87590463956139070015727376229, 4.08379331213660696936032823688, 4.59546122557385217629789820771, 4.65168210636875656123279438289, 4.69007782614118770680168609932, 5.26162929618539286342752890034, 5.45725681129531527842857408575, 5.51353532999741307869215834485, 5.90508806644719562779067785606, 6.00036035500522940460046212826, 6.10369930287206406779904534040, 6.36512830745111480021899452796, 6.45943846897056164066551919052, 6.99326595092749575480987268386