| L(s) = 1 | + 6·19-s − 48·37-s + 16·64-s − 66·73-s − 12·109-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 | + 1.37·19-s − 7.89·37-s + 2·64-s − 7.72·73-s − 1.14·109-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.056713256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.056713256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2} \) |
| 5 | \( 1 + 236 T^{6} + 40071 T^{12} + 236 p^{6} T^{18} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 34 T^{3} + 813 T^{6} - 34 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 11 | \( 1 + 1712 T^{6} + 1159383 T^{12} + 1712 p^{6} T^{18} + p^{12} T^{24} \) |
| 13 | \( ( 1 + 38 T^{3} - 753 T^{6} + 38 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 20 T^{2} + 111 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 19 | \( ( 1 - 8 T + p T^{2} )^{6}( 1 + 7 T + p T^{2} )^{6} \) |
| 23 | \( 1 - 520 T^{6} - 147765489 T^{12} - 520 p^{6} T^{18} + p^{12} T^{24} \) |
| 29 | \( 1 + 46478 T^{6} + 1565381163 T^{12} + 46478 p^{6} T^{18} + p^{12} T^{24} \) |
| 31 | \( ( 1 + 92 T^{3} - 21327 T^{6} + 92 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{6} \) |
| 41 | \( 1 + 97382 T^{6} + 4733149683 T^{12} + 97382 p^{6} T^{18} + p^{12} T^{24} \) |
| 43 | \( ( 1 - 88 T^{3} - 71763 T^{6} - 88 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 47 | \( 1 - 13246 T^{6} - 10603758813 T^{12} - 13246 p^{6} T^{18} + p^{12} T^{24} \) |
| 53 | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{6} \) |
| 59 | \( 1 - 235312 T^{6} + 13191203703 T^{12} - 235312 p^{6} T^{18} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 790 T^{3} + 397119 T^{6} - 790 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 1064 T^{3} + 831333 T^{6} + 1064 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 88 T^{2} + 2703 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{3} \) |
| 73 | \( ( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{6} \) |
| 79 | \( ( 1 + 1316 T^{3} + 1238817 T^{6} + 1316 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 326576 T^{6} - 220288489593 T^{12} + 326576 p^{6} T^{18} + p^{12} T^{24} \) |
| 89 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{6} \) |
| 97 | \( ( 1 + 1694 T^{3} + 1956963 T^{6} + 1694 p^{3} T^{9} + p^{6} T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.38820118557286355417146479891, −3.34856335420024081147512409483, −3.30087905215984719612479392837, −3.27034683558264249916914133659, −2.91097444701345467779875168704, −2.87677162167449744720379689589, −2.70296862158360008463576048151, −2.61337457262061862131272820235, −2.55284369076679863162709418970, −2.54071501279500209673692587599, −2.46559426982897407613324545775, −2.12438216014932638184035308650, −2.06588318468725458854096480770, −1.91186421346557494617224878107, −1.80280562456634389175073435232, −1.67425066860703962612759841841, −1.58199364083437925803951477050, −1.43767813133883439299441713143, −1.43262748222905769819619402922, −1.32740546626422994784308053203, −1.09503061048420658879406261007, −0.828688624491273752995699474596, −0.47832071229389434722840464672, −0.39185498606565115252436813155, −0.13300248356930204438894034282,
0.13300248356930204438894034282, 0.39185498606565115252436813155, 0.47832071229389434722840464672, 0.828688624491273752995699474596, 1.09503061048420658879406261007, 1.32740546626422994784308053203, 1.43262748222905769819619402922, 1.43767813133883439299441713143, 1.58199364083437925803951477050, 1.67425066860703962612759841841, 1.80280562456634389175073435232, 1.91186421346557494617224878107, 2.06588318468725458854096480770, 2.12438216014932638184035308650, 2.46559426982897407613324545775, 2.54071501279500209673692587599, 2.55284369076679863162709418970, 2.61337457262061862131272820235, 2.70296862158360008463576048151, 2.87677162167449744720379689589, 2.91097444701345467779875168704, 3.27034683558264249916914133659, 3.30087905215984719612479392837, 3.34856335420024081147512409483, 3.38820118557286355417146479891
Plot not available for L-functions of degree greater than 10.
