| L(s) = 1 | − 8·7-s − 18·9-s + 144·23-s + 204·41-s + 312·47-s − 156·49-s + 144·63-s + 81·81-s − 132·89-s + 40·103-s − 430·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 1.15e3·161-s + 163-s + 167-s + 500·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 8/7·7-s − 2·9-s + 6.26·23-s + 4.97·41-s + 6.63·47-s − 3.18·49-s + 16/7·63-s + 81-s − 1.48·89-s + 0.388·103-s − 3.55·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 7.15·161-s + 0.00613·163-s + 0.00598·167-s + 2.95·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.626732333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.626732333\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 215 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 250 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 137 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 359 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 36 T + p^{2} T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 710 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 898 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2630 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 51 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 1346 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 5186 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6530 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2650 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 3935 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1982 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 3769 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 12286 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12455 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 13342 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−6.51180681576582328474458087612, −6.22472249816235434784411608849, −6.05018132161954007608390866122, −5.93494791710646696223097914597, −5.48384772761678136886571849247, −5.45441963648880500125073354701, −5.39274378208477642486595068223, −4.98560567228931137092397503629, −4.96011895861127147697800885804, −4.35857177325058513628331682317, −4.25711102796303231124866333225, −4.23120262007545870377423690291, −3.73575667446045280392624767811, −3.48333700089001818476563177279, −3.13851181419352781654782191829, −2.98831914032550043844350886306, −2.70538231569053635761434208779, −2.64988788896926623477879709963, −2.55788318047844756868153076385, −2.25401661545185603183118307952, −1.35919306731854215441587787777, −1.17326730825798765935022751775, −0.876219774651551619949570503031, −0.76959585641357547277770510435, −0.23968010648174564964369898628,
0.23968010648174564964369898628, 0.76959585641357547277770510435, 0.876219774651551619949570503031, 1.17326730825798765935022751775, 1.35919306731854215441587787777, 2.25401661545185603183118307952, 2.55788318047844756868153076385, 2.64988788896926623477879709963, 2.70538231569053635761434208779, 2.98831914032550043844350886306, 3.13851181419352781654782191829, 3.48333700089001818476563177279, 3.73575667446045280392624767811, 4.23120262007545870377423690291, 4.25711102796303231124866333225, 4.35857177325058513628331682317, 4.96011895861127147697800885804, 4.98560567228931137092397503629, 5.39274378208477642486595068223, 5.45441963648880500125073354701, 5.48384772761678136886571849247, 5.93494791710646696223097914597, 6.05018132161954007608390866122, 6.22472249816235434784411608849, 6.51180681576582328474458087612
