| L(s) = 1 | − 132·9-s + 1.12e3·11-s + 2.00e3·19-s − 2.68e3·29-s − 4.49e3·31-s + 4.61e4·41-s + 5.65e4·49-s − 1.25e5·59-s + 2.82e4·61-s + 9.44e4·71-s + 1.31e5·79-s − 3.07e4·81-s + 1.10e5·89-s − 1.47e5·99-s + 2.05e5·101-s − 5.57e4·109-s + 1.76e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.73e5·169-s + ⋯ |
| L(s) = 1 | − 0.543·9-s + 2.79·11-s + 1.27·19-s − 0.591·29-s − 0.840·31-s + 4.28·41-s + 3.36·49-s − 4.68·59-s + 0.970·61-s + 2.22·71-s + 2.37·79-s − 0.520·81-s + 1.47·89-s − 1.51·99-s + 2.00·101-s − 0.449·109-s + 1.09·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.00·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.247163853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.247163853\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 44 p T^{2} + 5350 p^{2} T^{4} + 44 p^{11} T^{6} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 8084 p T^{2} + 1352943558 T^{4} - 8084 p^{11} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 560 T + 381926 T^{2} - 560 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 373292 T^{2} + 167403787638 T^{4} - 373292 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 1613316 T^{2} + 4602934743878 T^{4} + 1613316 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 1000 T + 4533462 T^{2} - 1000 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 7126092 T^{2} + 48612883261958 T^{4} - 7126092 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1340 T - 8758306 T^{2} + 1340 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2248 T + 57017022 T^{2} + 2248 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 211585036 T^{2} + 19959790722550422 T^{4} - 211585036 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 23076 T + 352280470 T^{2} - 23076 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 313073372 T^{2} + 49182412950657078 T^{4} - 313073372 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 104863596 T^{2} + 104529727880710502 T^{4} - 104863596 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1357205900 T^{2} + 805869805574922198 T^{4} - 1357205900 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 62584 T + 2277965606 T^{2} + 62584 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 14108 T + 1110042462 T^{2} - 14108 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1448611388 T^{2} + 3060385933795078038 T^{4} - 1448611388 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 47208 T + 4011779662 T^{2} - 47208 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4251788572 T^{2} + 9098112686809466598 T^{4} - 4251788572 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 65904 T + 3994274078 T^{2} - 65904 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9324010812 T^{2} + 49682906641812448598 T^{4} - 9324010812 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 55020 T + 10818978262 T^{2} - 55020 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1419021116 T^{2} - 92174956617487206138 T^{4} - 1419021116 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−8.288077925443334931152754554731, −7.57902096414777838489711309524, −7.52987174481569512291311133334, −7.48236443857403381000736574194, −7.42184940528152505578619169180, −6.55987717897669483023177860546, −6.52832107721148282458173637386, −6.33107349795434463400165443388, −5.92447573148231228392136259668, −5.83140873929610993660679064165, −5.44062763867067363440672798555, −4.93873971678306543111978895773, −4.88442844167355178674293189297, −4.25264503609791318396816609494, −3.91729743966997393741856810305, −3.87193098101024648137916353650, −3.66963154977059156917830317509, −2.92030834639149969110573835161, −2.89989297697243845827381778246, −2.15199276393246822627597818817, −2.08299394151947769604299972687, −1.37675173839207154554255800829, −1.01336140216721744538601795209, −0.879002499092536039593285989187, −0.39479657103774302475497443343,
0.39479657103774302475497443343, 0.879002499092536039593285989187, 1.01336140216721744538601795209, 1.37675173839207154554255800829, 2.08299394151947769604299972687, 2.15199276393246822627597818817, 2.89989297697243845827381778246, 2.92030834639149969110573835161, 3.66963154977059156917830317509, 3.87193098101024648137916353650, 3.91729743966997393741856810305, 4.25264503609791318396816609494, 4.88442844167355178674293189297, 4.93873971678306543111978895773, 5.44062763867067363440672798555, 5.83140873929610993660679064165, 5.92447573148231228392136259668, 6.33107349795434463400165443388, 6.52832107721148282458173637386, 6.55987717897669483023177860546, 7.42184940528152505578619169180, 7.48236443857403381000736574194, 7.52987174481569512291311133334, 7.57902096414777838489711309524, 8.288077925443334931152754554731
