| L(s) = 1 | − 8·7-s + 8·9-s + 32·25-s + 12·49-s − 64·63-s + 80·79-s + 30·81-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s − 256·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 8/3·9-s + 32/5·25-s + 12/7·49-s − 8.06·63-s + 9.00·79-s + 10/3·81-s − 3.63·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 4.92·169-s + 0.0760·173-s − 19.3·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1540994193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1540994193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| good | 5 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + p T^{2} )^{8} \) |
| 19 | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p T^{2} )^{8} \) |
| 59 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 10 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.77555314275788396650016768619, −4.49845730701127049438178197958, −4.16688667075670683940645121724, −4.12381235344667461555755829206, −3.98978945481440810663671448963, −3.74142265716364723103502605634, −3.71336544659971089523523125935, −3.63446049877766505589082165917, −3.45620235271945572286410989286, −3.27706998711262453755857357083, −3.22206064920503753287867575028, −3.00066929480712735276241387986, −2.98885799164017745486805745354, −2.57869259744024151210033242552, −2.50174007538239128358724425673, −2.39613972916846851806105338939, −2.35205801354651184525454605043, −2.16455526880373444119123555656, −1.57542542556234187223751856720, −1.53384758518308385213584364139, −1.28255966979167215658691597815, −1.02504051934004328024292856779, −0.919538965353800595283517205868, −0.884436993660180715737190786705, −0.05436564044930796610488716938,
0.05436564044930796610488716938, 0.884436993660180715737190786705, 0.919538965353800595283517205868, 1.02504051934004328024292856779, 1.28255966979167215658691597815, 1.53384758518308385213584364139, 1.57542542556234187223751856720, 2.16455526880373444119123555656, 2.35205801354651184525454605043, 2.39613972916846851806105338939, 2.50174007538239128358724425673, 2.57869259744024151210033242552, 2.98885799164017745486805745354, 3.00066929480712735276241387986, 3.22206064920503753287867575028, 3.27706998711262453755857357083, 3.45620235271945572286410989286, 3.63446049877766505589082165917, 3.71336544659971089523523125935, 3.74142265716364723103502605634, 3.98978945481440810663671448963, 4.12381235344667461555755829206, 4.16688667075670683940645121724, 4.49845730701127049438178197958, 4.77555314275788396650016768619
Plot not available for L-functions of degree greater than 10.
