| L(s) = 1 | − 2·9-s + 8·13-s + 16-s − 12·17-s − 8·25-s + 24·29-s + 16·31-s − 12·43-s − 24·47-s − 24·53-s + 8·61-s + 16·67-s − 12·73-s + 9·81-s − 24·83-s + 36·89-s − 24·97-s + 48·107-s − 16·117-s + 20·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.21·13-s + 1/4·16-s − 2.91·17-s − 8/5·25-s + 4.45·29-s + 2.87·31-s − 1.82·43-s − 3.50·47-s − 3.29·53-s + 1.02·61-s + 1.95·67-s − 1.40·73-s + 81-s − 2.63·83-s + 3.81·89-s − 2.43·97-s + 4.64·107-s − 1.47·117-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/6·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01665573003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01665573003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T^{4} + T^{8} \) |
| 3 | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 73 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 8 T^{2} + 72 T^{3} - 313 T^{4} + 72 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 12 T + 72 T^{2} + 120 T^{3} - 1006 T^{4} - 8316 T^{5} - 20160 T^{6} + 40068 T^{7} + 459795 T^{8} + 40068 p T^{9} - 20160 p^{2} T^{10} - 8316 p^{3} T^{11} - 1006 p^{4} T^{12} + 120 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 311 T^{4} - 183120 T^{8} - 311 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 12 T + 2 p T^{2} - 336 T^{3} + 2379 T^{4} - 336 p T^{5} + 2 p^{3} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 1487 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 6 T + 18 T^{2} - 408 T^{3} - 3073 T^{4} - 408 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 24 T + 288 T^{2} + 1392 T^{3} - 5122 T^{4} - 123048 T^{5} - 509184 T^{6} + 3281400 T^{7} + 49848195 T^{8} + 3281400 p T^{9} - 509184 p^{2} T^{10} - 123048 p^{3} T^{11} - 5122 p^{4} T^{12} + 1392 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 24 T + 288 T^{2} + 960 T^{3} - 14191 T^{4} - 219312 T^{5} - 715680 T^{6} + 8653464 T^{7} + 122283120 T^{8} + 8653464 p T^{9} - 715680 p^{2} T^{10} - 219312 p^{3} T^{11} - 14191 p^{4} T^{12} + 960 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 86 T^{2} + 3915 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T - 92 T^{2} + 56 T^{3} + 6967 T^{4} + 56 p T^{5} - 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} + 816 T^{3} - 7753 T^{4} + 816 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 8 T^{2} - 8882 T^{4} + 128 p T^{6} + 10771 p^{2} T^{8} + 128 p^{3} T^{10} - 8882 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 + 12 T + 72 T^{2} + 2136 T^{3} + 23074 T^{4} + 153972 T^{5} + 2467584 T^{6} + 22482180 T^{7} + 131711475 T^{8} + 22482180 p T^{9} + 2467584 p^{2} T^{10} + 153972 p^{3} T^{11} + 23074 p^{4} T^{12} + 2136 p^{5} T^{13} + 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 164 T^{2} + 8842 T^{4} + 913808 T^{6} + 119138899 T^{8} + 913808 p^{2} T^{10} + 8842 p^{4} T^{12} + 164 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 18 T + 73 T^{2} - 1314 T^{3} + 26244 T^{4} - 1314 p T^{5} + 73 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 24 T + 288 T^{2} + 528 T^{3} - 27746 T^{4} - 407016 T^{5} - 1638144 T^{6} + 17385480 T^{7} + 355712835 T^{8} + 17385480 p T^{9} - 1638144 p^{2} T^{10} - 407016 p^{3} T^{11} - 27746 p^{4} T^{12} + 528 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.69634876173574438483501196084, −4.59720766166789643355947523515, −4.46336719055544335414211939246, −4.45730492142474983027025772248, −4.27786450608560827561813851672, −3.90230789665957531607012414790, −3.68313008965972859989831785004, −3.65736881544745113415374770343, −3.62647776918225261861986306600, −3.40718163615597987106115357064, −3.40159227801415237635013026074, −3.03660337321870352529225728313, −2.82278030401197105094768688030, −2.80620779561508277825173392752, −2.59634503024644314094165416158, −2.58512831988562123357800604425, −2.27224191956580121758689906776, −1.99606648072367873337238392905, −1.88301477340091183149176225935, −1.69270354872466454592118022580, −1.33936069611921193826091572537, −1.32538630403546745845260842274, −0.855580680546214935999229866788, −0.820289303098806223278579690597, −0.02049543277553428425949683741,
0.02049543277553428425949683741, 0.820289303098806223278579690597, 0.855580680546214935999229866788, 1.32538630403546745845260842274, 1.33936069611921193826091572537, 1.69270354872466454592118022580, 1.88301477340091183149176225935, 1.99606648072367873337238392905, 2.27224191956580121758689906776, 2.58512831988562123357800604425, 2.59634503024644314094165416158, 2.80620779561508277825173392752, 2.82278030401197105094768688030, 3.03660337321870352529225728313, 3.40159227801415237635013026074, 3.40718163615597987106115357064, 3.62647776918225261861986306600, 3.65736881544745113415374770343, 3.68313008965972859989831785004, 3.90230789665957531607012414790, 4.27786450608560827561813851672, 4.45730492142474983027025772248, 4.46336719055544335414211939246, 4.59720766166789643355947523515, 4.69634876173574438483501196084
Plot not available for L-functions of degree greater than 10.
