| L(s) = 1 | + 16·5-s + 48·13-s − 16·17-s + 24·25-s + 80·29-s − 16·37-s − 80·41-s + 152·49-s − 176·53-s − 272·61-s + 768·65-s − 16·73-s − 256·85-s + 240·89-s + 400·97-s + 528·101-s + 560·109-s − 336·113-s + 520·121-s − 1.04e3·125-s + 127-s + 131-s + 137-s + 139-s + 1.28e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 16/5·5-s + 3.69·13-s − 0.941·17-s + 0.959·25-s + 2.75·29-s − 0.432·37-s − 1.95·41-s + 3.10·49-s − 3.32·53-s − 4.45·61-s + 11.8·65-s − 0.219·73-s − 3.01·85-s + 2.69·89-s + 4.12·97-s + 5.22·101-s + 5.13·109-s − 2.97·113-s + 4.29·121-s − 8.31·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 8.82·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(51.06874211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(51.06874211\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T + 84 T^{2} - 472 T^{3} + 2822 T^{4} - 472 p^{2} T^{5} + 84 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 7 | \( 1 - 152 T^{2} + 1476 p T^{4} - 514600 T^{6} + 25187654 T^{8} - 514600 p^{4} T^{10} + 1476 p^{9} T^{12} - 152 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 - 260 T^{2} + 40038 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 24 T + 540 T^{2} - 5736 T^{3} + 89702 T^{4} - 5736 p^{2} T^{5} + 540 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 8 T + 252 T^{2} - 2888 T^{3} + 48902 T^{4} - 2888 p^{2} T^{5} + 252 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( 1 - 1224 T^{2} + 1027804 T^{4} - 554589432 T^{6} + 236077316358 T^{8} - 554589432 p^{4} T^{10} + 1027804 p^{8} T^{12} - 1224 p^{12} T^{14} + p^{16} T^{16} \) |
| 23 | \( 1 - 3272 T^{2} + 5007132 T^{4} - 4715977336 T^{6} + 5669682278 p^{2} T^{8} - 4715977336 p^{4} T^{10} + 5007132 p^{8} T^{12} - 3272 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( ( 1 - 40 T + 2196 T^{2} - 31160 T^{3} + 1501766 T^{4} - 31160 p^{2} T^{5} + 2196 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( 1 - 4248 T^{2} + 9652444 T^{4} - 14917063464 T^{6} + 16677690731718 T^{8} - 14917063464 p^{4} T^{10} + 9652444 p^{8} T^{12} - 4248 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( ( 1 + 8 T + 1980 T^{2} - 54920 T^{3} + 1119206 T^{4} - 54920 p^{2} T^{5} + 1980 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 40 T + 2556 T^{2} - 43240 T^{3} - 289594 T^{4} - 43240 p^{2} T^{5} + 2556 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 - 5960 T^{2} + 23041116 T^{4} - 59814611320 T^{6} + 126481252740230 T^{8} - 59814611320 p^{4} T^{10} + 23041116 p^{8} T^{12} - 5960 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( 1 - 12360 T^{2} + 70380700 T^{4} - 250403346936 T^{6} + 638353123484742 T^{8} - 250403346936 p^{4} T^{10} + 70380700 p^{8} T^{12} - 12360 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 + 88 T + 10068 T^{2} + 520520 T^{3} + 37632134 T^{4} + 520520 p^{2} T^{5} + 10068 p^{4} T^{6} + 88 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 - 7688 T^{2} + 37931484 T^{4} - 124184601784 T^{6} + 382130398823558 T^{8} - 124184601784 p^{4} T^{10} + 37931484 p^{8} T^{12} - 7688 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 136 T + 14076 T^{2} + 1001336 T^{3} + 63819302 T^{4} + 1001336 p^{2} T^{5} + 14076 p^{4} T^{6} + 136 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8644 T^{2} + 58097190 T^{4} - 8644 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 18120 T^{2} + 152088604 T^{4} - 903092412024 T^{6} + 4737261672397254 T^{8} - 903092412024 p^{4} T^{10} + 152088604 p^{8} T^{12} - 18120 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 8 T + 5212 T^{2} + 653240 T^{3} - 3523322 T^{4} + 653240 p^{2} T^{5} + 5212 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 26776 T^{2} + 333200860 T^{4} - 2815930225960 T^{6} + 19175645821896646 T^{8} - 2815930225960 p^{4} T^{10} + 333200860 p^{8} T^{12} - 26776 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 - 15752 T^{2} + 70662492 T^{4} - 102795674680 T^{6} + 590797586843654 T^{8} - 102795674680 p^{4} T^{10} + 70662492 p^{8} T^{12} - 15752 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 120 T + 30300 T^{2} - 2699976 T^{3} + 352873862 T^{4} - 2699976 p^{2} T^{5} + 30300 p^{4} T^{6} - 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 - 200 T + 21276 T^{2} - 3009400 T^{3} + 385130822 T^{4} - 3009400 p^{2} T^{5} + 21276 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.87189455896222145922695881319, −3.71895122270549003774386757779, −3.60825548764279053912255958660, −3.52994784304783173949107384581, −3.43068744541868047413079277700, −3.40880543912367347158888244083, −3.19807614943447079427327145096, −3.05500284880032950598103946401, −2.87407526258103776122664793623, −2.66314584492414232532814750361, −2.53096459426691874059985662017, −2.50493107587629151424830985074, −2.08575261727129289088870215889, −2.04188521461815392989575473338, −1.88383463618709827547959354556, −1.82969459802608028995248714661, −1.80352598872074198115575507500, −1.65787183287388615058910534514, −1.43035711326549308185042646561, −1.25320702331859140843354923092, −1.09160195838025041039676382210, −0.71427050997182289445619412069, −0.67739469477376889910939236780, −0.40017707799363025640272824257, −0.35957479168079409958517716425,
0.35957479168079409958517716425, 0.40017707799363025640272824257, 0.67739469477376889910939236780, 0.71427050997182289445619412069, 1.09160195838025041039676382210, 1.25320702331859140843354923092, 1.43035711326549308185042646561, 1.65787183287388615058910534514, 1.80352598872074198115575507500, 1.82969459802608028995248714661, 1.88383463618709827547959354556, 2.04188521461815392989575473338, 2.08575261727129289088870215889, 2.50493107587629151424830985074, 2.53096459426691874059985662017, 2.66314584492414232532814750361, 2.87407526258103776122664793623, 3.05500284880032950598103946401, 3.19807614943447079427327145096, 3.40880543912367347158888244083, 3.43068744541868047413079277700, 3.52994784304783173949107384581, 3.60825548764279053912255958660, 3.71895122270549003774386757779, 3.87189455896222145922695881319
Plot not available for L-functions of degree greater than 10.
