| L(s) = 1 | − 4·4-s − 4·9-s + 18·16-s + 24·25-s + 16·36-s − 16·37-s + 32·43-s − 64·64-s + 48·67-s − 16·79-s + 14·81-s − 96·100-s − 64·109-s − 64·121-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s − 128·172-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4/3·9-s + 9/2·16-s + 24/5·25-s + 8/3·36-s − 2.63·37-s + 4.87·43-s − 8·64-s + 5.86·67-s − 1.80·79-s + 14/9·81-s − 9.59·100-s − 6.13·109-s − 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s − 9.75·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224590710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224590710\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 4 T^{2} + 2 T^{4} - 16 T^{6} - 29 T^{8} - 16 p^{2} T^{10} + 2 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( 1 \) |
| good | 2 | \( ( 1 + p T^{2} - 3 T^{4} - p T^{6} + 17 T^{8} - p^{3} T^{10} - 3 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} )^{2} \) |
| 5 | \( ( 1 - 12 T^{2} + 12 p T^{4} - 408 T^{6} + 2831 T^{8} - 408 p^{2} T^{10} + 12 p^{5} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 + 32 T^{2} + 534 T^{4} + 7936 T^{6} + 101555 T^{8} + 7936 p^{2} T^{10} + 534 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 32 T^{2} + 496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - 52 T^{2} + 1500 T^{4} - 32552 T^{6} + 584639 T^{8} - 32552 p^{2} T^{10} + 1500 p^{4} T^{12} - 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 + 44 T^{2} + 858 T^{4} + 15664 T^{6} + 332867 T^{8} + 15664 p^{2} T^{10} + 858 p^{4} T^{12} + 44 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 + 72 T^{2} + 2902 T^{4} + 88128 T^{6} + 2185347 T^{8} + 88128 p^{2} T^{10} + 2902 p^{4} T^{12} + 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 32 T^{2} + 1546 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 31 | \( ( 1 + 116 T^{2} + 8178 T^{4} + 389296 T^{6} + 14002547 T^{8} + 389296 p^{2} T^{10} + 8178 p^{4} T^{12} + 116 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 4 T - 12 T^{2} - 184 T^{3} - 1177 T^{4} - 184 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 108 T^{2} + 6180 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - 2 T + p T^{2} )^{16} \) |
| 47 | \( ( 1 - 12 T^{2} + 3378 T^{4} + 91824 T^{6} + 5260979 T^{8} + 91824 p^{2} T^{10} + 3378 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 + 172 T^{2} + 16858 T^{4} + 1222576 T^{6} + 70959139 T^{8} + 1222576 p^{2} T^{10} + 16858 p^{4} T^{12} + 172 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 - 108 T^{2} + 2298 T^{4} - 259632 T^{6} + 35002211 T^{8} - 259632 p^{2} T^{10} + 2298 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 + 128 T^{2} + 4848 T^{4} + 524032 T^{6} + 58268591 T^{8} + 524032 p^{2} T^{10} + 4848 p^{4} T^{12} + 128 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 12 T + 46 T^{2} + 432 T^{3} - 4533 T^{4} + 432 p T^{5} + 46 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 200 T^{2} + 19690 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 73 | \( ( 1 + 144 T^{2} + 5952 T^{4} + 594144 T^{6} + 79666271 T^{8} + 594144 p^{2} T^{10} + 5952 p^{4} T^{12} + 144 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 4 T - 138 T^{2} - 16 T^{3} + 16211 T^{4} - 16 p T^{5} - 138 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 108 T^{2} + 6894 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 89 | \( ( 1 - 180 T^{2} + 8796 T^{4} - 1397160 T^{6} + 237766175 T^{8} - 1397160 p^{2} T^{10} + 8796 p^{4} T^{12} - 180 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 144 T^{2} + 19200 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
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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
−4.05037845792766085612569917134, −3.91546672918464874938179210756, −3.88359049212381138630418940609, −3.75838323334632605064940076582, −3.45975111307209341283861582349, −3.28843307348142896242758116767, −3.27230094310359044999942581245, −3.26125756001051966891932867216, −3.07137122995004186946605742686, −3.05695956599226385659574534741, −3.02244202048940261832472350360, −2.83609241608523228049818539441, −2.66824267021723260529103175473, −2.66570875939531358337415340659, −2.54837363360199736271229335815, −2.36428661541490663536266079636, −2.09452973834173561412889798452, −2.00713822380037392047829450371, −1.95795393577682332276490748576, −1.53080854492373331605397733171, −1.41993778272846412382261628574, −1.34354831200758456703061749938, −0.856411079747160462686738896653, −0.855945004991916102711359088139, −0.72612498428476433537522632040,
0.72612498428476433537522632040, 0.855945004991916102711359088139, 0.856411079747160462686738896653, 1.34354831200758456703061749938, 1.41993778272846412382261628574, 1.53080854492373331605397733171, 1.95795393577682332276490748576, 2.00713822380037392047829450371, 2.09452973834173561412889798452, 2.36428661541490663536266079636, 2.54837363360199736271229335815, 2.66570875939531358337415340659, 2.66824267021723260529103175473, 2.83609241608523228049818539441, 3.02244202048940261832472350360, 3.05695956599226385659574534741, 3.07137122995004186946605742686, 3.26125756001051966891932867216, 3.27230094310359044999942581245, 3.28843307348142896242758116767, 3.45975111307209341283861582349, 3.75838323334632605064940076582, 3.88359049212381138630418940609, 3.91546672918464874938179210756, 4.05037845792766085612569917134
Plot not available for L-functions of degree greater than 10.
