| L(s) = 1 | + 3·4-s − 10·9-s + 5·16-s − 12·19-s − 30·36-s + 22·49-s − 60·59-s + 24·61-s − 6·64-s + 12·71-s − 36·76-s + 48·79-s + 43·81-s + 48·89-s − 36·101-s − 30·121-s + 127-s + 131-s + 137-s + 139-s − 50·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 3.33·9-s + 5/4·16-s − 2.75·19-s − 5·36-s + 22/7·49-s − 7.81·59-s + 3.07·61-s − 3/4·64-s + 1.42·71-s − 4.12·76-s + 5.40·79-s + 43/9·81-s + 5.08·89-s − 3.58·101-s − 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 4.16·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4440080620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4440080620\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 13 | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 2 | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2}( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 3 | \( ( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 15 T^{2} + 104 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 13 T^{2} - 120 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 37 T^{2} + 528 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T^{2} + 1182 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 11 T^{2} - 1248 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 75 T^{2} + 3944 T^{4} + 75 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2}( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 + 108 T^{2} + 5990 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 152 T^{2} + 10638 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 30 T + 465 T^{2} + 4950 T^{3} + 41444 T^{4} + 4950 p T^{5} + 465 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 7 T^{2} - 180 T^{3} + 6264 T^{4} - 180 p T^{5} + 7 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 46 T^{2} + 4705 T^{4} + 532082 T^{6} - 22946924 T^{8} + 532082 p^{2} T^{10} + 4705 p^{4} T^{12} - 46 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 6 T + 129 T^{2} - 702 T^{3} + 9500 T^{4} - 702 p T^{5} + 129 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p T^{2} )^{8} \) |
| 79 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 + 168 T^{2} + 18734 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 24 T + 411 T^{2} - 5256 T^{3} + 57128 T^{4} - 5256 p T^{5} + 411 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 - 238 T^{2} + 26689 T^{4} - 2650606 T^{6} + 275809348 T^{8} - 2650606 p^{2} T^{10} + 26689 p^{4} T^{12} - 238 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−7.18865746680236474386755340585, −6.88972277598566996675735052984, −6.63245746008387717216145898156, −6.55411531417451381422625263340, −6.50253374052466875181641935995, −6.41429547456782242924466642019, −6.07842581584673251710227019933, −6.02045540426939672321457339109, −5.77957670439194024130564255707, −5.71215355939499884915873279929, −5.37045689866446928364770460358, −5.06021038345619471715466873224, −5.02036978095841196206089201887, −4.90972446654755545152386280136, −4.45173881883099361984777950509, −4.16201228896303372136314573058, −3.99893193282605464660449145949, −3.59127200802506070753869368518, −3.48460614171024375361892633897, −3.20822318418770939066641300386, −2.71184241147790884934468388573, −2.67516543151767637733438152719, −2.38666506630074288429616164973, −2.19809918440530383914969316923, −1.69638152110125240897701556347,
1.69638152110125240897701556347, 2.19809918440530383914969316923, 2.38666506630074288429616164973, 2.67516543151767637733438152719, 2.71184241147790884934468388573, 3.20822318418770939066641300386, 3.48460614171024375361892633897, 3.59127200802506070753869368518, 3.99893193282605464660449145949, 4.16201228896303372136314573058, 4.45173881883099361984777950509, 4.90972446654755545152386280136, 5.02036978095841196206089201887, 5.06021038345619471715466873224, 5.37045689866446928364770460358, 5.71215355939499884915873279929, 5.77957670439194024130564255707, 6.02045540426939672321457339109, 6.07842581584673251710227019933, 6.41429547456782242924466642019, 6.50253374052466875181641935995, 6.55411531417451381422625263340, 6.63245746008387717216145898156, 6.88972277598566996675735052984, 7.18865746680236474386755340585
Plot not available for L-functions of degree greater than 10.
