| L(s) = 1 | + 8·13-s + 16·25-s − 32·37-s + 8·49-s + 16·61-s + 32·73-s + 32·97-s + 8·109-s − 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 68·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.21·13-s + 16/5·25-s − 5.26·37-s + 8/7·49-s + 2.04·61-s + 3.74·73-s + 3.24·97-s + 0.766·109-s − 1.45·121-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 − 5.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.170821962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.170821962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 4 T^{2} - 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 56 T^{2} + 1335 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 32 T^{2} + 1263 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 8 T + 63 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 44 T^{2} + 1014 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 92 T^{2} + 5190 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 4 T + 99 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 100 T^{2} + 10506 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 32 T^{2} - 5214 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 8 T + 135 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 292 T^{2} + 33690 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 128 T^{2} + 19263 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.99557070560224783081753651921, −3.97339442783899826064389072727, −3.73745749403307057825780352593, −3.58494987320886144053407202266, −3.58130597224761691897949914836, −3.54428132894404643631857636815, −3.48995617241475416343943260040, −3.26698610727380333379902549762, −3.19032840110733127513842602433, −2.91098131368860403564471886859, −2.65975586666681119550042584927, −2.61656845367979174074258994556, −2.59928521179697943938234306164, −2.40706715761136472726768092491, −2.15980085712322007580241192679, −2.07089597361678431584136503355, −1.80848605431782376790338662490, −1.63853021297597360293966384768, −1.49084228384264020194594214846, −1.36276344377961105051424729666, −1.29067306132049335121077859207, −0.798419385698732741281378316009, −0.795097190894943578618211415835, −0.62636944169353104747156097358, −0.26070986207376858384835213848,
0.26070986207376858384835213848, 0.62636944169353104747156097358, 0.795097190894943578618211415835, 0.798419385698732741281378316009, 1.29067306132049335121077859207, 1.36276344377961105051424729666, 1.49084228384264020194594214846, 1.63853021297597360293966384768, 1.80848605431782376790338662490, 2.07089597361678431584136503355, 2.15980085712322007580241192679, 2.40706715761136472726768092491, 2.59928521179697943938234306164, 2.61656845367979174074258994556, 2.65975586666681119550042584927, 2.91098131368860403564471886859, 3.19032840110733127513842602433, 3.26698610727380333379902549762, 3.48995617241475416343943260040, 3.54428132894404643631857636815, 3.58130597224761691897949914836, 3.58494987320886144053407202266, 3.73745749403307057825780352593, 3.97339442783899826064389072727, 3.99557070560224783081753651921
Plot not available for L-functions of degree greater than 10.
