| L(s) = 1 | − 4·9-s + 24·41-s + 24·49-s − 26·81-s − 8·89-s + 68·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 3.74·41-s + 24/7·49-s − 2.88·81-s − 0.847·89-s + 6.18·121-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 + 4.30·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.09079710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.09079710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−3.59695025774478556589644183123, −3.55773992333957888777873800859, −3.27227736939251968638041781582, −3.07667112476848635233065951633, −3.02810112660903401058232020019, −3.00894460282619592593678535657, −2.86648039038340014433134073054, −2.85842460885069053354522728851, −2.65050762315209851037048858378, −2.58471902532718894014288085888, −2.33440120821207507857513306277, −2.29379610390489729861795386611, −2.16422400914050609095237668200, −2.04834286343848006680247694264, −1.93499989224102457799805911149, −1.64753644880678847841600648491, −1.60966168125923872046140163629, −1.47279546757400289306531069159, −1.17977551937094503185373322435, −1.14720553700608517191676820822, −0.814400254076622232425883088918, −0.70516200040660348069073472688, −0.51024498801620632162235455627, −0.41998611454189192793924161490, −0.36406028847228352297394652487,
0.36406028847228352297394652487, 0.41998611454189192793924161490, 0.51024498801620632162235455627, 0.70516200040660348069073472688, 0.814400254076622232425883088918, 1.14720553700608517191676820822, 1.17977551937094503185373322435, 1.47279546757400289306531069159, 1.60966168125923872046140163629, 1.64753644880678847841600648491, 1.93499989224102457799805911149, 2.04834286343848006680247694264, 2.16422400914050609095237668200, 2.29379610390489729861795386611, 2.33440120821207507857513306277, 2.58471902532718894014288085888, 2.65050762315209851037048858378, 2.85842460885069053354522728851, 2.86648039038340014433134073054, 3.00894460282619592593678535657, 3.02810112660903401058232020019, 3.07667112476848635233065951633, 3.27227736939251968638041781582, 3.55773992333957888777873800859, 3.59695025774478556589644183123
Plot not available for L-functions of degree greater than 10.
