| L(s) = 1 | + 4·3-s − 4·7-s + 6·9-s − 4·13-s + 4·17-s − 16·21-s − 12·23-s − 4·27-s + 8·29-s + 12·37-s − 16·39-s + 4·43-s − 20·47-s + 8·49-s + 16·51-s − 12·53-s − 16·59-s + 24·61-s − 24·63-s + 12·67-s − 48·69-s − 4·73-s − 37·81-s + 4·83-s + 32·87-s − 40·89-s + 16·91-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.51·7-s + 2·9-s − 1.10·13-s + 0.970·17-s − 3.49·21-s − 2.50·23-s − 0.769·27-s + 1.48·29-s + 1.97·37-s − 2.56·39-s + 0.609·43-s − 2.91·47-s + 8/7·49-s + 2.24·51-s − 1.64·53-s − 2.08·59-s + 3.07·61-s − 3.02·63-s + 1.46·67-s − 5.77·69-s − 0.468·73-s − 4.11·81-s + 0.439·83-s + 3.43·87-s − 4.23·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8778060241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8778060241\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 194 T^{4} - 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2542 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 468 T^{3} + 3038 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 164 T^{3} + 3358 T^{4} - 164 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1860 T^{3} + 15182 T^{4} + 1860 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} + 180 T^{3} - 6274 T^{4} + 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 44 T^{3} - 3602 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 196 T^{3} + 3646 T^{4} - 196 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.02558053219048857656282060789, −6.89920405406967857558751139335, −6.54481875889106710777271262470, −6.33564638108919453824947338459, −6.05220381751170670190642923638, −5.97054122278278134411115808651, −5.73737725336631536026890676842, −5.35142498583258395507090100484, −5.33568015802922598492884484575, −4.71666115217938112836416365514, −4.57654636027014894737182158983, −4.40663893044425551023444704375, −4.10495002767833472807711009460, −3.78024266868612202801429035816, −3.68193822661456719619718423470, −3.20564327021690093252040719150, −3.08923233077835281193056801137, −2.98750177986397782142118324957, −2.80775495534921349368565919216, −2.27278774160508165359373902784, −2.06332024960361941121247990895, −2.05413978950888941734754582229, −1.41800298114524329292012137314, −0.942111914166941289948394546811, −0.16171795013699758381330114852,
0.16171795013699758381330114852, 0.942111914166941289948394546811, 1.41800298114524329292012137314, 2.05413978950888941734754582229, 2.06332024960361941121247990895, 2.27278774160508165359373902784, 2.80775495534921349368565919216, 2.98750177986397782142118324957, 3.08923233077835281193056801137, 3.20564327021690093252040719150, 3.68193822661456719619718423470, 3.78024266868612202801429035816, 4.10495002767833472807711009460, 4.40663893044425551023444704375, 4.57654636027014894737182158983, 4.71666115217938112836416365514, 5.33568015802922598492884484575, 5.35142498583258395507090100484, 5.73737725336631536026890676842, 5.97054122278278134411115808651, 6.05220381751170670190642923638, 6.33564638108919453824947338459, 6.54481875889106710777271262470, 6.89920405406967857558751139335, 7.02558053219048857656282060789
