| L(s) = 1 | − 12·9-s − 24·49-s + 48·79-s + 64·81-s − 48·109-s + 112·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 64·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
| L(s) = 1 | − 4·9-s − 3.42·49-s + 5.40·79-s + 64/9·81-s − 4.59·109-s + 10.1·121-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 + 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.742897635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.742897635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + 2 p T^{2} + 22 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| good | 11 | \( ( 1 - 28 T^{2} + 393 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 13 | \( ( 1 + 16 T^{2} + 82 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 46 T^{2} + 1062 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 19 | \( ( 1 - 46 T^{2} + 1126 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 23 | \( ( 1 + 12 T^{2} - 31 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 29 | \( ( 1 - 12 T^{2} + 1313 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 - 94 T^{2} + 4006 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 37 | \( ( 1 - 69 T^{2} + p^{2} T^{4} )^{8} \) |
| 41 | \( ( 1 + 74 T^{2} + 4606 T^{4} + 74 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 82 T^{2} + 3379 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 34 T^{2} + 1582 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 53 | \( ( 1 + 32 T^{2} + 5374 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 59 | \( ( 1 + 76 T^{2} + 3906 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 61 | \( ( 1 - 94 T^{2} + 6526 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 67 | \( ( 1 - 89 T^{2} + p^{2} T^{4} )^{8} \) |
| 71 | \( ( 1 - 164 T^{2} + 13681 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 - 6 T + 147 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 - 242 T^{2} + 28294 T^{4} - 242 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 + 116 T^{2} + 6706 T^{4} + 116 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 + 294 T^{2} + 38222 T^{4} + 294 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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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
−2.27396748441911710114949009966, −2.18468683051140167739567715796, −2.17866908796482221687907986444, −2.16841918407169936774808876932, −2.12428122809745659505676058358, −1.93660646134244854150906319141, −1.82558457525625264826089904949, −1.70240336770601719909401502969, −1.59571861921925315417867728883, −1.52005351641915515026291037903, −1.49294930571329548489046929163, −1.41248044732470887417366995020, −1.36961983446613691502164983657, −1.36805373955684371342528493067, −1.29607572722679199767805864398, −1.14115140487251036010074380555, −1.00295794850219311341446362886, −0.60459466585064258833363547519, −0.57492350888771121942729988801, −0.53502261239925637779723489293, −0.51292630124451714316640098298, −0.51212319758058525101640820975, −0.46016141250308945801669735405, −0.39751348776079812312291763400, −0.10337673759872324789265239776,
0.10337673759872324789265239776, 0.39751348776079812312291763400, 0.46016141250308945801669735405, 0.51212319758058525101640820975, 0.51292630124451714316640098298, 0.53502261239925637779723489293, 0.57492350888771121942729988801, 0.60459466585064258833363547519, 1.00295794850219311341446362886, 1.14115140487251036010074380555, 1.29607572722679199767805864398, 1.36805373955684371342528493067, 1.36961983446613691502164983657, 1.41248044732470887417366995020, 1.49294930571329548489046929163, 1.52005351641915515026291037903, 1.59571861921925315417867728883, 1.70240336770601719909401502969, 1.82558457525625264826089904949, 1.93660646134244854150906319141, 2.12428122809745659505676058358, 2.16841918407169936774808876932, 2.17866908796482221687907986444, 2.18468683051140167739567715796, 2.27396748441911710114949009966
Plot not available for L-functions of degree greater than 10.
