| L(s) = 1 | − 2-s − 3-s + 4·5-s + 6-s + 2·7-s − 4·10-s − 4·13-s − 2·14-s − 4·15-s + 2·17-s − 2·21-s − 24·23-s + 5·25-s + 4·26-s − 10·29-s + 4·30-s + 8·31-s + 32-s − 2·34-s + 8·35-s + 2·37-s + 4·39-s − 2·41-s + 2·42-s + 16·43-s + 24·46-s + 2·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1.78·5-s + 0.408·6-s + 0.755·7-s − 1.26·10-s − 1.10·13-s − 0.534·14-s − 1.03·15-s + 0.485·17-s − 0.436·21-s − 5.00·23-s + 25-s + 0.784·26-s − 1.85·29-s + 0.730·30-s + 1.43·31-s + 0.176·32-s − 0.342·34-s + 1.35·35-s + 0.328·37-s + 0.640·39-s − 0.312·41-s + 0.308·42-s + 2.43·43-s + 3.53·46-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333055385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333055385\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | | \( 1 \) |
| good | 5 | $C_4\times C_2$ | \( 1 - 4 T + 11 T^{2} - 24 T^{3} + 41 T^{4} - 24 p T^{5} + 11 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 2 T - 3 T^{2} + 20 T^{3} - 19 T^{4} + 20 p T^{5} - 3 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 40 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 60 p T^{5} - 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 420 T^{3} + 2141 T^{4} + 420 p T^{5} + 71 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 16 p T^{5} + 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_4\times C_2$ | \( 1 - 2 T - 33 T^{2} + 140 T^{3} + 941 T^{4} + 140 p T^{5} - 33 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4\times C_2$ | \( 1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 156 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 47 | $C_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 180 T^{3} + 1661 T^{4} + 180 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_4\times C_2$ | \( 1 + 4 T - 37 T^{2} - 360 T^{3} + 521 T^{4} - 360 p T^{5} - 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - 8 T + 3 T^{2} + 464 T^{3} - 3895 T^{4} + 464 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 2 T - 67 T^{2} - 276 T^{3} + 4205 T^{4} - 276 p T^{5} - 67 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 6 T - 37 T^{2} + 660 T^{3} - 1259 T^{4} + 660 p T^{5} - 37 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 10 T + 21 T^{2} - 580 T^{3} - 7459 T^{4} - 580 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 + 4 T - 67 T^{2} - 600 T^{3} + 3161 T^{4} - 600 p T^{5} - 67 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - 2 T - 93 T^{2} + 380 T^{3} + 8261 T^{4} + 380 p T^{5} - 93 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.59475428932325568562321877020, −7.51051349777525205982037910712, −7.20100803779847773133243923953, −6.64779194400396546638654646058, −6.45150368271735882527899088121, −6.07639380052804603894934918786, −5.87660365934706755878391445563, −5.87152262261866208464348368764, −5.85731919396513069878508898824, −5.61861061913938010939195723787, −5.24000129418750066977189733016, −4.65260034158783467260069562989, −4.50686399533060329943275600364, −4.45586196387025145687576414119, −4.33993096844994034404597729975, −3.64696916839387534200561762328, −3.57249201402569499327021840399, −3.05317059375928460858040905301, −2.76733570904848788870478417095, −2.09849354589634995346058730785, −2.06700088088945674123126100375, −1.89891918537028913759514080020, −1.78785858697309912090724360907, −0.829690035986764433367018455699, −0.43071074627931510176201845945,
0.43071074627931510176201845945, 0.829690035986764433367018455699, 1.78785858697309912090724360907, 1.89891918537028913759514080020, 2.06700088088945674123126100375, 2.09849354589634995346058730785, 2.76733570904848788870478417095, 3.05317059375928460858040905301, 3.57249201402569499327021840399, 3.64696916839387534200561762328, 4.33993096844994034404597729975, 4.45586196387025145687576414119, 4.50686399533060329943275600364, 4.65260034158783467260069562989, 5.24000129418750066977189733016, 5.61861061913938010939195723787, 5.85731919396513069878508898824, 5.87152262261866208464348368764, 5.87660365934706755878391445563, 6.07639380052804603894934918786, 6.45150368271735882527899088121, 6.64779194400396546638654646058, 7.20100803779847773133243923953, 7.51051349777525205982037910712, 7.59475428932325568562321877020
