| L(s) = 1 | − 6·4-s − 6·9-s + 13·16-s + 36·36-s − 6·64-s + 9·81-s + 72·121-s + 127-s + 131-s + 137-s + 139-s − 78·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 3·4-s − 2·9-s + 13/4·16-s + 6·36-s − 3/4·64-s + 81-s + 6.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.5·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04501877681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04501877681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 + 3 T^{2} + 7 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 2 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 36 T^{2} + 553 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 36 T^{2} + 2113 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 4370 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 204 T^{2} + 17353 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 70 T^{2} + 1179 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 204 T^{2} + 18913 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 13730 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 276 T^{2} + 33313 T^{4} + 276 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
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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
−4.63551374383456508573874680378, −4.52932241960333453520760884076, −4.45430129627352214445097803791, −4.42866599104787517010859463672, −4.41399924860492681751392714137, −4.38246814417981268494422555232, −3.83117910821008392603431391658, −3.71926510659350114498748403702, −3.63309509335693622623564456967, −3.58277553856310040786373012310, −3.46883369951088156917398547757, −3.33376386521510354470825606681, −2.96382707438719468846422912008, −2.82222404903425104582719977440, −2.76756259704859255798247468964, −2.46243456803328184106975842712, −2.42853375116889831060229058189, −2.21504355745170351523548726950, −1.90558844655801470196211457479, −1.69938318644832911656625623526, −1.43735324962412166409103779611, −1.03288671532338169759142507312, −0.902265000529904177501658699656, −0.43033674998247893913707154931, −0.082825186433060653568919060284,
0.082825186433060653568919060284, 0.43033674998247893913707154931, 0.902265000529904177501658699656, 1.03288671532338169759142507312, 1.43735324962412166409103779611, 1.69938318644832911656625623526, 1.90558844655801470196211457479, 2.21504355745170351523548726950, 2.42853375116889831060229058189, 2.46243456803328184106975842712, 2.76756259704859255798247468964, 2.82222404903425104582719977440, 2.96382707438719468846422912008, 3.33376386521510354470825606681, 3.46883369951088156917398547757, 3.58277553856310040786373012310, 3.63309509335693622623564456967, 3.71926510659350114498748403702, 3.83117910821008392603431391658, 4.38246814417981268494422555232, 4.41399924860492681751392714137, 4.42866599104787517010859463672, 4.45430129627352214445097803791, 4.52932241960333453520760884076, 4.63551374383456508573874680378
Plot not available for L-functions of degree greater than 10.
