L(s) = 1 | − 4·5-s − 4·9-s − 4·13-s − 24·17-s + 6·25-s − 4·29-s − 24·37-s + 24·41-s + 16·45-s + 20·49-s − 32·53-s − 8·61-s + 16·65-s − 16·73-s + 9·81-s + 96·85-s − 72·89-s − 24·97-s + 24·101-s + 48·113-s + 16·117-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 4/3·9-s − 1.10·13-s − 5.82·17-s + 6/5·25-s − 0.742·29-s − 3.94·37-s + 3.74·41-s + 2.38·45-s + 20/7·49-s − 4.39·53-s − 1.02·61-s + 1.98·65-s − 1.87·73-s + 81-s + 10.4·85-s − 7.63·89-s − 2.43·97-s + 2.38·101-s + 4.51·113-s + 1.47·117-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06075210534\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06075210534\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T^{2} + 7 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( ( 1 + T + T^{2} )^{4} \) |
good | 7 | \( 1 - 20 T^{2} + 31 p T^{4} - 1700 T^{6} + 11488 T^{8} - 1700 p^{2} T^{10} + 31 p^{5} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 16 T^{2} + 135 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T - 8 T^{2} - 28 T^{3} - 77 T^{4} - 28 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( 1 - 84 T^{2} + 4249 T^{4} - 146916 T^{6} + 3866784 T^{8} - 146916 p^{2} T^{10} + 4249 p^{4} T^{12} - 84 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 2 T + 5 T^{2} - 118 T^{3} - 956 T^{4} - 118 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 32 T^{2} - 194 T^{4} + 22528 T^{6} - 424061 T^{8} + 22528 p^{2} T^{10} - 194 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 - 80 T^{2} + 4551 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 140 T^{2} + 10417 T^{4} - 667100 T^{6} + 36151408 T^{8} - 667100 p^{2} T^{10} + 10417 p^{4} T^{12} - 140 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 59 | \( 1 - 204 T^{2} + 24490 T^{4} - 2073456 T^{6} + 136317219 T^{8} - 2073456 p^{2} T^{10} + 24490 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 4 T - 95 T^{2} - 44 T^{3} + 7624 T^{4} - 44 p T^{5} - 95 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 20 T^{2} + 2257 T^{4} - 216700 T^{6} - 18498272 T^{8} - 216700 p^{2} T^{10} + 2257 p^{4} T^{12} + 20 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 132 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 2 T^{2} - 6237 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 76 T^{2} - 8231 T^{4} + 17404 T^{6} + 108946864 T^{8} + 17404 p^{2} T^{10} - 8231 p^{4} T^{12} + 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 18 T + 199 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 12 T - 26 T^{2} - 288 T^{3} + 5523 T^{4} - 288 p T^{5} - 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.04873735305136271954241857959, −4.01356569402469990347116658201, −3.92054567331883457243546804739, −3.75987814417553559238394910258, −3.64489062492511113299615876563, −3.41417029521922794731860479887, −3.09234699597166224377129512529, −3.05696734341137304727106070670, −2.93057643499232815587457039140, −2.89809705435619363359998763902, −2.81700044605552682955604445337, −2.68122863968921272943040881154, −2.35915732856958396086625064798, −2.30915337811698363747287333715, −2.30409148068877064700445689992, −2.02431205784507597977831943530, −1.90909187132106449779668422261, −1.69412408142830060879929082305, −1.46870259109114233442460401339, −1.38790881880210903600317056402, −1.29331881539908759541850336964, −0.69199618928708724499951044134, −0.39602432871546571969236509593, −0.16505397072170838678348247836, −0.14603365943198417206469598043,
0.14603365943198417206469598043, 0.16505397072170838678348247836, 0.39602432871546571969236509593, 0.69199618928708724499951044134, 1.29331881539908759541850336964, 1.38790881880210903600317056402, 1.46870259109114233442460401339, 1.69412408142830060879929082305, 1.90909187132106449779668422261, 2.02431205784507597977831943530, 2.30409148068877064700445689992, 2.30915337811698363747287333715, 2.35915732856958396086625064798, 2.68122863968921272943040881154, 2.81700044605552682955604445337, 2.89809705435619363359998763902, 2.93057643499232815587457039140, 3.05696734341137304727106070670, 3.09234699597166224377129512529, 3.41417029521922794731860479887, 3.64489062492511113299615876563, 3.75987814417553559238394910258, 3.92054567331883457243546804739, 4.01356569402469990347116658201, 4.04873735305136271954241857959
Plot not available for L-functions of degree greater than 10.