| L(s) = 1 | − 3·4-s + 9-s − 12·11-s + 4·16-s + 8·19-s + 16·29-s − 3·36-s + 36·44-s − 24·49-s − 8·59-s + 4·61-s + 16·71-s − 24·76-s + 18·89-s − 12·99-s − 52·101-s − 52·109-s − 48·116-s + 98·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1/3·9-s − 3.61·11-s + 16-s + 1.83·19-s + 2.97·29-s − 1/2·36-s + 5.42·44-s − 3.42·49-s − 1.04·59-s + 0.512·61-s + 1.89·71-s − 2.75·76-s + 1.90·89-s − 1.20·99-s − 5.17·101-s − 4.98·109-s − 4.45·116-s + 8.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515648859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515648859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3 T^{2} + 5 T^{4} + 3 T^{6} - 11 T^{8} + 3 p^{2} T^{10} + 5 p^{4} T^{12} + 3 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 6 T + 5 T^{2} - 6 T^{3} + 49 T^{4} - 6 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 - 18 T^{2} + 155 T^{4} - 3708 T^{6} + 76189 T^{8} - 3708 p^{2} T^{10} + 155 p^{4} T^{12} - 18 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 + 30 T^{2} + 251 T^{4} - 5580 T^{6} - 204659 T^{8} - 5580 p^{2} T^{10} + 251 p^{4} T^{12} + 30 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 62 p T^{5} - 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 + 26 T^{2} + 147 T^{4} - 9932 T^{6} - 335995 T^{8} - 9932 p^{2} T^{10} + 147 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 8 T + 5 T^{2} + 8 p T^{3} - 59 p T^{4} + 8 p^{2} T^{5} + 5 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 99 T^{2} + 2307 T^{4} - 114263 T^{6} - 8470320 T^{8} - 114263 p^{2} T^{10} + 2307 p^{4} T^{12} + 99 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 31 T^{2} + 180 T^{3} + 1501 T^{4} + 180 p T^{5} - 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 80 T^{2} + 5118 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 50 T^{2} + 4731 T^{4} + 330760 T^{6} + 12156101 T^{8} + 330760 p^{2} T^{10} + 4731 p^{4} T^{12} + 50 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 111 T^{2} + 6987 T^{4} + 278333 T^{6} + 12828480 T^{8} + 278333 p^{2} T^{10} + 6987 p^{4} T^{12} + 111 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 4 T - 43 T^{2} - 408 T^{3} + 905 T^{4} - 408 p T^{5} - 43 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 64 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 90 T^{2} + 10451 T^{4} + 1055640 T^{6} + 61563541 T^{8} + 1055640 p^{2} T^{10} + 10451 p^{4} T^{12} + 90 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 46 T^{2} + 8787 T^{4} + 723068 T^{6} + 37435205 T^{8} + 723068 p^{2} T^{10} + 8787 p^{4} T^{12} + 46 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 242 T^{2} + 15195 T^{4} - 1254908 T^{6} - 234429451 T^{8} - 1254908 p^{2} T^{10} + 15195 p^{4} T^{12} + 242 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 9 T - 43 T^{2} + 993 T^{3} - 4460 T^{4} + 993 p T^{5} - 43 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 70 T^{2} + 17931 T^{4} + 2275580 T^{6} + 161028101 T^{8} + 2275580 p^{2} T^{10} + 17931 p^{4} T^{12} + 70 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−5.09044398739612876746806182670, −4.93433021605731431232315414194, −4.82589463347140738956402801571, −4.64075556118722235473432436070, −4.47845220995514335667623601175, −4.24488227509477733320519367732, −4.22157174662066579333343165409, −4.20490672303743677765162090944, −4.18644551030163123263051204546, −3.52571972829921336727536224199, −3.39269652150154048344834074429, −3.36478494992875225967949960631, −3.09686955299249117027453417222, −3.09146569225188333250764417132, −2.81439220715274740579872616090, −2.80114168250959235066036832150, −2.78058268904171661920998505644, −2.24787356005077656924115650226, −2.07367995342400592765986513129, −1.80438698724136430299997971061, −1.74005083121128260080562046207, −1.41471885095522751821427263707, −0.803430060524521198873504055207, −0.62304614053055206472985788463, −0.47264429947674406341020691398,
0.47264429947674406341020691398, 0.62304614053055206472985788463, 0.803430060524521198873504055207, 1.41471885095522751821427263707, 1.74005083121128260080562046207, 1.80438698724136430299997971061, 2.07367995342400592765986513129, 2.24787356005077656924115650226, 2.78058268904171661920998505644, 2.80114168250959235066036832150, 2.81439220715274740579872616090, 3.09146569225188333250764417132, 3.09686955299249117027453417222, 3.36478494992875225967949960631, 3.39269652150154048344834074429, 3.52571972829921336727536224199, 4.18644551030163123263051204546, 4.20490672303743677765162090944, 4.22157174662066579333343165409, 4.24488227509477733320519367732, 4.47845220995514335667623601175, 4.64075556118722235473432436070, 4.82589463347140738956402801571, 4.93433021605731431232315414194, 5.09044398739612876746806182670
Plot not available for L-functions of degree greater than 10.
