| L(s) = 1 | − 4·43-s − 10·49-s − 44·73-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 0.609·43-s − 1.42·49-s − 5.14·73-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 + 8·169-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 + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.252190751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.252190751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( ( 1 - p T^{2} )^{4} \) |
| good | 5 | \( 1 - 31 T^{4} + 336 T^{8} - 31 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 233 T^{4} + 39648 T^{8} + 233 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - p T^{2} )^{8} \) |
| 17 | \( 1 + 353 T^{4} + 41088 T^{8} + 353 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 158 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 + p T^{2} )^{8} \) |
| 43 | \( ( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 - 1207 T^{4} - 3422832 T^{8} - 1207 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - 103 T^{2} + 6888 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{8} \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - p T^{2} )^{8} \) |
| 83 | \( ( 1 - 5678 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 - p T^{2} )^{8} \) |
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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
−3.56152556315113691443071490765, −3.51935319289925344322605803965, −3.50452761164406480218324392579, −3.44674017738261844730865851085, −3.18057909434203012996303894704, −3.05466612529565233358387486831, −2.92039920803672267573712557934, −2.89011427827274298587508320605, −2.71806484957337314713512538682, −2.60928584283827658675864979218, −2.39331544031075826617341648637, −2.30402878489543813586160321173, −2.23655524628980052958865653295, −2.12223927202963058311630871797, −1.82173209835503165593003817328, −1.67370350017004768666386418413, −1.66447033400597866886376969994, −1.46689295672020673130002087480, −1.28837201177492236081854725319, −1.26898684025014324042785367295, −0.937823593254205181878926434019, −0.920245764446407294774929266868, −0.40813769632161091863707039184, −0.34154935130658166619040734392, −0.21922955850372402693366741329,
0.21922955850372402693366741329, 0.34154935130658166619040734392, 0.40813769632161091863707039184, 0.920245764446407294774929266868, 0.937823593254205181878926434019, 1.26898684025014324042785367295, 1.28837201177492236081854725319, 1.46689295672020673130002087480, 1.66447033400597866886376969994, 1.67370350017004768666386418413, 1.82173209835503165593003817328, 2.12223927202963058311630871797, 2.23655524628980052958865653295, 2.30402878489543813586160321173, 2.39331544031075826617341648637, 2.60928584283827658675864979218, 2.71806484957337314713512538682, 2.89011427827274298587508320605, 2.92039920803672267573712557934, 3.05466612529565233358387486831, 3.18057909434203012996303894704, 3.44674017738261844730865851085, 3.50452761164406480218324392579, 3.51935319289925344322605803965, 3.56152556315113691443071490765
Plot not available for L-functions of degree greater than 10.
