| L(s) = 1 | + 24·9-s − 4·17-s + 18·25-s − 24·41-s − 10·49-s + 4·73-s + 324·81-s − 80·89-s + 72·97-s + 40·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + 163-s + 167-s + 16·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 8·9-s − 0.970·17-s + 18/5·25-s − 3.74·41-s − 1.42·49-s + 0.468·73-s + 36·81-s − 8.47·89-s + 7.31·97-s + 3.76·113-s + 2·121-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 − 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(19.66063051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.66063051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( ( 1 + T^{2} )^{4} \) |
| good | 3 | \( ( 1 - p T^{2} )^{8} \) |
| 5 | \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - p T^{2} + 144 T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 72 T^{2} + 2750 T^{4} - 72 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 80 T^{2} + 3294 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 56 T^{2} + 1470 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 59 T^{2} + 2160 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 165 T^{2} + 11096 T^{4} + 165 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 120 T^{2} + 10334 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 233 T^{2} + 21000 T^{4} - 233 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 152 T^{2} + 14526 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 64 T^{2} + 9054 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 132 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 128 T^{2} + 10878 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 132 T^{2} + 14486 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 10 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−4.09006766000337156255922678264, −3.97869620793611312304494692195, −3.96680065718833501099000171312, −3.82484768759651005478387558909, −3.79522109804007823105823398979, −3.52271547231018074195822402822, −3.48783627673262881550463045916, −3.31280331372246770376784802328, −3.24542229861721662217447194214, −2.86706302912044422922297189904, −2.82550531143838767342062149329, −2.80040268024149812142606011721, −2.32698484820637173328689697976, −2.29672218191831991215928374594, −2.04456391096378707103181693170, −2.00086097701687889257534285920, −1.80386150065715661652496502166, −1.67657201673385603884662542277, −1.47828214726312615245377401213, −1.41934589845176613582732728408, −1.16025064904092767914570652128, −1.09145196081869241164768145030, −1.07626457821753659168834057154, −0.68827073265550031454047029351, −0.30353064188474975068237736203,
0.30353064188474975068237736203, 0.68827073265550031454047029351, 1.07626457821753659168834057154, 1.09145196081869241164768145030, 1.16025064904092767914570652128, 1.41934589845176613582732728408, 1.47828214726312615245377401213, 1.67657201673385603884662542277, 1.80386150065715661652496502166, 2.00086097701687889257534285920, 2.04456391096378707103181693170, 2.29672218191831991215928374594, 2.32698484820637173328689697976, 2.80040268024149812142606011721, 2.82550531143838767342062149329, 2.86706302912044422922297189904, 3.24542229861721662217447194214, 3.31280331372246770376784802328, 3.48783627673262881550463045916, 3.52271547231018074195822402822, 3.79522109804007823105823398979, 3.82484768759651005478387558909, 3.96680065718833501099000171312, 3.97869620793611312304494692195, 4.09006766000337156255922678264
Plot not available for L-functions of degree greater than 10.
