L(s) = 1 | − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 4·4-s + 10·16-s + 12·23-s − 20·64-s + 81-s − 48·92-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9990037749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9990037749\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 - T^{4} + T^{8} \) |
| 7 | \( 1 \) |
good | 5 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 13 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 17 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 19 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 23 | \( ( 1 - T )^{8}( 1 - T + T^{2} )^{4} \) |
| 29 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 41 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 43 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 59 | \( ( 1 + T^{4} )^{4} \) |
| 61 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 67 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 71 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 73 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 89 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.76280385939169173187367722733, −3.50268629685441115190876341914, −3.45514841073596214632302704648, −3.33895036233477295349514455936, −3.23085070703288250942261952430, −3.22372426112599712290023322287, −3.21449256421386131795569253312, −3.08521346697873599964414926152, −3.08286166330424703047554455548, −2.73292651074023791419285258641, −2.72664569764473842005391873648, −2.52433929311177067181008101565, −2.34686414110017071426609793477, −2.31129795111952419094481376649, −2.24748951670814615707599313911, −1.78168855134543336327498769590, −1.54520793365046715443503950040, −1.50218452842280610816521720598, −1.19352398737308935427423036342, −1.16845926179826974997399715363, −1.08569701635602419120883950654, −0.948192522010199699720626754331, −0.907058511574132158144474429475, −0.870459814367495461448571515200, −0.34386840733286357329229250223,
0.34386840733286357329229250223, 0.870459814367495461448571515200, 0.907058511574132158144474429475, 0.948192522010199699720626754331, 1.08569701635602419120883950654, 1.16845926179826974997399715363, 1.19352398737308935427423036342, 1.50218452842280610816521720598, 1.54520793365046715443503950040, 1.78168855134543336327498769590, 2.24748951670814615707599313911, 2.31129795111952419094481376649, 2.34686414110017071426609793477, 2.52433929311177067181008101565, 2.72664569764473842005391873648, 2.73292651074023791419285258641, 3.08286166330424703047554455548, 3.08521346697873599964414926152, 3.21449256421386131795569253312, 3.22372426112599712290023322287, 3.23085070703288250942261952430, 3.33895036233477295349514455936, 3.45514841073596214632302704648, 3.50268629685441115190876341914, 3.76280385939169173187367722733
Plot not available for L-functions of degree greater than 10.