| L(s) = 1 | + 8·4-s + 16·16-s + 40·49-s − 464·61-s − 128·64-s + 176·109-s − 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 296·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 320·196-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 2·4-s + 16-s + 0.816·49-s − 7.60·61-s − 2·64-s + 1.61·109-s − 0.727·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.75·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 1.63·196-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9747930868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9747930868\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{4} \) |
| 11 | \( ( 1 + 2 p T^{2} + p^{4} T^{4} )^{4} \) |
| 13 | \( ( 1 + 74 T^{2} + p^{4} T^{4} )^{4} \) |
| 17 | \( ( 1 + 470 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{4} \) |
| 29 | \( ( 1 + 1594 T^{2} + p^{4} T^{4} )^{4} \) |
| 31 | \( ( 1 - 722 T^{2} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 362 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3010 T^{2} + p^{4} T^{4} )^{4} \) |
| 43 | \( ( 1 - 2290 T^{2} + p^{4} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4402 T^{2} + p^{4} T^{4} )^{4} \) |
| 53 | \( ( 1 + 4646 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 6698 T^{2} + p^{4} T^{4} )^{4} \) |
| 61 | \( ( 1 + 58 T + p^{2} T^{2} )^{8} \) |
| 67 | \( ( 1 - 8626 T^{2} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 578 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 + p^{2} T^{2} )^{8} \) |
| 79 | \( ( 1 - 12434 T^{2} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 12754 T^{2} + p^{4} T^{4} )^{4} \) |
| 89 | \( ( 1 + 10210 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 - 7582 T^{2} + p^{4} T^{4} )^{4} \) |
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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
−4.23115149411477215314986009165, −4.15598856772392684449435119292, −3.66452623068792147569955768069, −3.64434795606554861693129735083, −3.61928543401930516145821813282, −3.34940781639855689042821517675, −3.21406249644019628782353713513, −3.11313371252978320591083048547, −3.03968488204979578917817062730, −2.86206316615390629165482248896, −2.66083287222179523702288028847, −2.58692435322790517796885784188, −2.44152210845263512338200475081, −2.40847751356842859859471874214, −1.98039305668382624059394230539, −1.94499346405417908425041416515, −1.70072472925701155522409016995, −1.63198134108980306597595108688, −1.58141507839193229443048122395, −1.36994823315192124338686124477, −1.17030307327030360346963497515, −0.78322721863248376204471309282, −0.60884530781620071184431401574, −0.39796159555150166963935411005, −0.06338959303676016404450528236,
0.06338959303676016404450528236, 0.39796159555150166963935411005, 0.60884530781620071184431401574, 0.78322721863248376204471309282, 1.17030307327030360346963497515, 1.36994823315192124338686124477, 1.58141507839193229443048122395, 1.63198134108980306597595108688, 1.70072472925701155522409016995, 1.94499346405417908425041416515, 1.98039305668382624059394230539, 2.40847751356842859859471874214, 2.44152210845263512338200475081, 2.58692435322790517796885784188, 2.66083287222179523702288028847, 2.86206316615390629165482248896, 3.03968488204979578917817062730, 3.11313371252978320591083048547, 3.21406249644019628782353713513, 3.34940781639855689042821517675, 3.61928543401930516145821813282, 3.64434795606554861693129735083, 3.66452623068792147569955768069, 4.15598856772392684449435119292, 4.23115149411477215314986009165
Plot not available for L-functions of degree greater than 10.
