| L(s) = 1 | − 32·23-s − 8·25-s + 32·47-s + 32·49-s − 16·73-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 88·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 6.67·23-s − 8/5·25-s + 4.66·47-s + 32/7·49-s − 1.87·73-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.147642698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.147642698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 6 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 12 T^{2} + 86 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 16 T^{2} + 450 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 100 T^{2} + 4134 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 48 T^{2} + 1730 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 92 T^{2} + 4086 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 80 T^{2} + 3234 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 68 T^{2} + 2886 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 92 T^{2} + 6486 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 76 T^{2} + 3510 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 144 T^{2} + 10754 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 108 T^{2} + 11894 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 176 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.49027242633389246384577752074, −3.48183720833441408415175927401, −3.11183012207828037755460216097, −3.08215663377109517789617094475, −2.91747419775219121432566737920, −2.69303644806048028269275114905, −2.64035817862989574170645809919, −2.58182130175816972508222906906, −2.44341945117512545622693417520, −2.43815750364308761691232386403, −2.19432588027343533872410625465, −2.07080780358699591398240267343, −2.01044225700902454194655090794, −1.93580496640464399459144092628, −1.91403918565696764511681619031, −1.63070533472860143077543687437, −1.44831241921127841171835712560, −1.44189028679844307138481424848, −1.28331082261708989100057329270, −0.886247768339204182859505759294, −0.73944934511652530176613158690, −0.72931809753088815101584231175, −0.41976598624303263365841756882, −0.40671177807319142099650355954, −0.14661823328838266141411561057,
0.14661823328838266141411561057, 0.40671177807319142099650355954, 0.41976598624303263365841756882, 0.72931809753088815101584231175, 0.73944934511652530176613158690, 0.886247768339204182859505759294, 1.28331082261708989100057329270, 1.44189028679844307138481424848, 1.44831241921127841171835712560, 1.63070533472860143077543687437, 1.91403918565696764511681619031, 1.93580496640464399459144092628, 2.01044225700902454194655090794, 2.07080780358699591398240267343, 2.19432588027343533872410625465, 2.43815750364308761691232386403, 2.44341945117512545622693417520, 2.58182130175816972508222906906, 2.64035817862989574170645809919, 2.69303644806048028269275114905, 2.91747419775219121432566737920, 3.08215663377109517789617094475, 3.11183012207828037755460216097, 3.48183720833441408415175927401, 3.49027242633389246384577752074
Plot not available for L-functions of degree greater than 10.
