| L(s) = 1 | + 4·11-s − 8·19-s + 18·29-s − 16·31-s + 24·41-s + 5·49-s − 24·59-s + 26·61-s − 12·71-s − 2·79-s + 6·89-s + 42·101-s + 10·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 1.83·19-s + 3.34·29-s − 2.87·31-s + 3.74·41-s + 5/7·49-s − 3.12·59-s + 3.32·61-s − 1.42·71-s − 0.225·79-s + 0.635·89-s + 4.17·101-s + 0.957·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.68253235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.68253235\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 27 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 53 T^{2} + 1233 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 4803 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2763 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 3273 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 133 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T - 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 221 T^{2} + 22233 T^{4} - 221 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + T - 99 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 13653 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 3 T + 175 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 233 T^{2} + 30873 T^{4} - 233 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.51244837548085123247581326151, −5.03041889105877330243689908506, −4.95012824678705333446608635528, −4.73901995725784631866277136249, −4.65528502929087056690134580392, −4.37710401466911563980523490988, −4.31322035495514626992767319081, −4.16827021884785451814499093148, −3.98533275039838462210827855752, −3.57770242654558117783104255617, −3.52233628927630057796817808312, −3.51232768243304598262609002093, −3.14214978771754881610802217527, −2.80429729725830376146695158658, −2.59346482903948967936327661477, −2.57470924060455460131522850736, −2.42109471644520601309162377921, −1.85107277238180022418646870874, −1.82093343494211743653927374425, −1.74671569369418801167875883130, −1.46852159883212616601801357912, −0.947880841513787772913800432476, −0.71961071004049629909059080833, −0.52579458505651217309570551696, −0.49801111023434256751069840332,
0.49801111023434256751069840332, 0.52579458505651217309570551696, 0.71961071004049629909059080833, 0.947880841513787772913800432476, 1.46852159883212616601801357912, 1.74671569369418801167875883130, 1.82093343494211743653927374425, 1.85107277238180022418646870874, 2.42109471644520601309162377921, 2.57470924060455460131522850736, 2.59346482903948967936327661477, 2.80429729725830376146695158658, 3.14214978771754881610802217527, 3.51232768243304598262609002093, 3.52233628927630057796817808312, 3.57770242654558117783104255617, 3.98533275039838462210827855752, 4.16827021884785451814499093148, 4.31322035495514626992767319081, 4.37710401466911563980523490988, 4.65528502929087056690134580392, 4.73901995725784631866277136249, 4.95012824678705333446608635528, 5.03041889105877330243689908506, 5.51244837548085123247581326151
