| L(s) = 1 | − 12·2-s + 78·4-s − 364·8-s + 1.36e3·16-s − 20·23-s − 4.36e3·32-s + 240·46-s − 14·49-s − 20·53-s + 1.23e4·64-s − 56·79-s − 4·81-s − 1.56e3·92-s + 168·98-s + 240·106-s − 32·107-s + 56·109-s − 24·113-s + 40·121-s + 127-s − 3.18e4·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 8.48·2-s + 39·4-s − 128.·8-s + 341.·16-s − 4.17·23-s − 772.·32-s + 35.3·46-s − 2·49-s − 2.74·53-s + 1.54e3·64-s − 6.30·79-s − 4/9·81-s − 162.·92-s + 16.9·98-s + 23.3·106-s − 3.09·107-s + 5.36·109-s − 2.25·113-s + 3.63·121-s + 0.0887·127-s − 2.81e3·128-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001357348413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001357348413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T )^{12} \) |
| 3 | \( 1 + 4 T^{4} + 10 p T^{6} + 4 p^{2} T^{8} + p^{6} T^{12} \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2 p T^{2} + 99 T^{4} + 652 T^{6} + 99 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| good | 11 | \( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 320 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 15 T^{2} + 43 p T^{4} + 5110 T^{6} + 43 p^{3} T^{8} + 15 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 29 T^{2} + 494 T^{4} - 5297 T^{6} + 494 p^{2} T^{8} - 29 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 4132 p^{2} T^{8} - 96 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 + 5 T + 49 T^{2} + 224 T^{3} + 49 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 29 | \( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 5535 p^{2} T^{8} - 109 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2623 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 70 T^{2} + 2907 T^{4} - 101660 T^{6} + 2907 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 3 T^{2} + 3166 T^{4} - 23 p T^{6} + 3166 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 209 T^{2} + 19659 T^{4} - 1076742 T^{6} + 19659 p^{2} T^{8} - 209 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 78 T^{2} + 7315 T^{4} - 322124 T^{6} + 7315 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 5 T + 95 T^{2} + 548 T^{3} + 95 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 59 | \( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 18587 p^{2} T^{8} + 179 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 3 p T^{2} + 12451 T^{4} - 609898 T^{6} + 12451 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 - 260 T^{2} + 33680 T^{4} - 2779070 T^{6} + 33680 p^{2} T^{8} - 260 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 21667 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 332 T^{2} + 52356 T^{4} + 4849846 T^{6} + 52356 p^{2} T^{8} + 332 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 14 T + 193 T^{2} + 2172 T^{3} + 193 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 - 309 T^{2} + 44338 T^{4} - 4238477 T^{6} + 44338 p^{2} T^{8} - 309 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 58036 p^{2} T^{8} + 348 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 158 T^{2} + 21375 T^{4} + 2986564 T^{6} + 21375 p^{2} T^{8} + 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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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
−2.86829857089241342578859911533, −2.83882082295117982151459478679, −2.78370733915314456408522491765, −2.71330614966276294001226399420, −2.67176842050924539014452602717, −2.57851234807687774706569265391, −2.36156316716072542746271582608, −2.20628937978825339825996412098, −2.09733954835952850192402001619, −1.96244441624210368675743243250, −1.93320622404050892587663754646, −1.92970360979325119435478835175, −1.92963280569703110967307646040, −1.56950214996938336022135897936, −1.51218222279076329315901243464, −1.49724285723152647848134786468, −1.45157294166072298863451466773, −1.28744276071301248021087525928, −1.12018324048485312692200148944, −0.889310530798013407224923511455, −0.74124230016518951995884508653, −0.67065405016812529948573236121, −0.34763995613045536531030086021, −0.11780701823042280933456622933, −0.094531384661864704082119839797,
0.094531384661864704082119839797, 0.11780701823042280933456622933, 0.34763995613045536531030086021, 0.67065405016812529948573236121, 0.74124230016518951995884508653, 0.889310530798013407224923511455, 1.12018324048485312692200148944, 1.28744276071301248021087525928, 1.45157294166072298863451466773, 1.49724285723152647848134786468, 1.51218222279076329315901243464, 1.56950214996938336022135897936, 1.92963280569703110967307646040, 1.92970360979325119435478835175, 1.93320622404050892587663754646, 1.96244441624210368675743243250, 2.09733954835952850192402001619, 2.20628937978825339825996412098, 2.36156316716072542746271582608, 2.57851234807687774706569265391, 2.67176842050924539014452602717, 2.71330614966276294001226399420, 2.78370733915314456408522491765, 2.83882082295117982151459478679, 2.86829857089241342578859911533
Plot not available for L-functions of degree greater than 10.
