| L(s) = 1 | − 14·16-s − 16·31-s + 88·61-s − 17·81-s − 88·121-s + 127-s + 131-s + 137-s + 139-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 + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 7/2·16-s − 2.87·31-s + 11.2·61-s − 1.88·81-s − 8·121-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 + 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 + 0.0655·233-s + 0.0646·239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.829828768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.829828768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 17 T^{4} + 208 T^{8} + 17 p^{4} T^{12} + p^{8} T^{16} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| good | 2 | \( ( 1 + 7 T^{4} + 33 T^{8} + 7 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 13 | \( ( 1 - 334 T^{4} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 + 382 T^{4} + 62403 T^{8} + 382 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 1057 T^{4} + 837408 T^{8} + 1057 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{8} \) |
| 31 | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{8} \) |
| 37 | \( ( 1 + 1294 T^{4} - 199725 T^{8} + 1294 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 41 | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{8} \) |
| 43 | \( ( 1 + 2543 T^{4} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 4222 T^{4} + 12945603 T^{8} + 4222 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 1778 T^{4} - 4729197 T^{8} - 1778 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 22 T^{2} - 2997 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{8} \) |
| 67 | \( ( 1 - 7151 T^{4} + 30985680 T^{8} - 7151 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 71 | \( ( 1 - 12 T + p T^{2} )^{8}( 1 + 12 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 3266 T^{4} - 17731485 T^{8} - 3266 p^{4} T^{12} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 154 T^{2} + 17475 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 12143 T^{4} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 143 T^{2} + 12528 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - 12094 T^{4} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.89803253190175519704652313501, −2.80563698804356765437694665974, −2.72850403269101540113904453940, −2.69907848730474511902543494076, −2.60772980841184607160244342440, −2.51303076505837124801440893747, −2.29437007544275936298546375319, −2.27698174141057516051954422903, −2.27290214327590416808685681161, −2.17111134602611000319221239002, −2.12452011533495633744865113450, −2.05075242819577907843712651699, −1.91808097319248163359675869163, −1.79254873964103804115361786399, −1.70384509049486757411128774348, −1.67128509491698987284023152262, −1.36490059584203482432231622161, −1.32370416983505126972841461959, −1.15755483147547056983736325371, −1.11120520840563786870833520059, −0.905691812102019486706154266872, −0.64803232059696220112041939579, −0.47983542113944709752270530694, −0.34049696756125568454004322635, −0.31940633012054008662272478736,
0.31940633012054008662272478736, 0.34049696756125568454004322635, 0.47983542113944709752270530694, 0.64803232059696220112041939579, 0.905691812102019486706154266872, 1.11120520840563786870833520059, 1.15755483147547056983736325371, 1.32370416983505126972841461959, 1.36490059584203482432231622161, 1.67128509491698987284023152262, 1.70384509049486757411128774348, 1.79254873964103804115361786399, 1.91808097319248163359675869163, 2.05075242819577907843712651699, 2.12452011533495633744865113450, 2.17111134602611000319221239002, 2.27290214327590416808685681161, 2.27698174141057516051954422903, 2.29437007544275936298546375319, 2.51303076505837124801440893747, 2.60772980841184607160244342440, 2.69907848730474511902543494076, 2.72850403269101540113904453940, 2.80563698804356765437694665974, 2.89803253190175519704652313501
Plot not available for L-functions of degree greater than 10.
