| L(s) = 1 | − 2·9-s + 36·17-s + 10·25-s − 4·49-s − 36·73-s + 19·81-s − 36·89-s − 96·113-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 72·153-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 8.73·17-s + 2·25-s − 4/7·49-s − 4.21·73-s + 19/9·81-s − 3.81·89-s − 9.03·113-s + 1.27·121-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 − 5.82·153-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.098517110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.098517110\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 + T^{2} - 8 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 - 7 T^{2} - 72 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 7 T^{2} - 312 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | \( ( 1 - 61 T^{2} + 2352 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 67 T^{2} + 2280 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 91 T^{2} + 5472 T^{4} + 91 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 113 T^{2} + 9288 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 77 T^{2} + 2208 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + T^{2} - 4488 T^{4} + p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2}( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.96280166596327501392016016270, −4.94484659759141918658929382263, −4.87246595385158032332613868729, −4.44854555028266885414207441560, −4.05102881319106145810637510002, −4.02865743936933805275183292511, −4.02455818587310813697000865993, −3.92110615410783594690340378474, −3.89810099323124256262399775153, −3.29207210449248555469820676671, −3.24316805434977761924415275522, −3.23081911973860292364268492628, −3.14913100296089988578281864624, −3.13041585379470395460645242294, −2.81131613405990301453412037535, −2.69807335596867841962478822600, −2.56323623286910209665701306601, −2.28154425166454555859248951814, −1.93867443373797405860210388279, −1.36312896730591228081064368954, −1.32778315975093716078252250746, −1.29159057992756876896181853998, −1.16610805740405323067524241190, −1.12716860289092540567033393810, −0.34666289553865611288580353702,
0.34666289553865611288580353702, 1.12716860289092540567033393810, 1.16610805740405323067524241190, 1.29159057992756876896181853998, 1.32778315975093716078252250746, 1.36312896730591228081064368954, 1.93867443373797405860210388279, 2.28154425166454555859248951814, 2.56323623286910209665701306601, 2.69807335596867841962478822600, 2.81131613405990301453412037535, 3.13041585379470395460645242294, 3.14913100296089988578281864624, 3.23081911973860292364268492628, 3.24316805434977761924415275522, 3.29207210449248555469820676671, 3.89810099323124256262399775153, 3.92110615410783594690340378474, 4.02455818587310813697000865993, 4.02865743936933805275183292511, 4.05102881319106145810637510002, 4.44854555028266885414207441560, 4.87246595385158032332613868729, 4.94484659759141918658929382263, 4.96280166596327501392016016270
Plot not available for L-functions of degree greater than 10.
