| L(s) = 1 | + 4·4-s + 3·16-s + 24·25-s + 8·37-s − 56·61-s − 12·64-s + 96·100-s − 80·109-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3/4·16-s + 24/5·25-s + 1.31·37-s − 7.17·61-s − 3/2·64-s + 48/5·100-s − 7.66·109-s − 6.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.38·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(21.47130071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(21.47130071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{2} + 13 T^{4} - 7 p^{2} T^{6} + 13 p^{2} T^{8} - p^{6} T^{10} + p^{6} T^{12} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 12 T^{2} + 19 p T^{4} - 568 T^{6} + 19 p^{3} T^{8} - 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 11 | \( ( 1 + 38 T^{2} + 65 p T^{4} + 9068 T^{6} + 65 p^{3} T^{8} + 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 11 T^{2} + 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{4} \) |
| 17 | \( ( 1 - 4 p T^{2} + 2167 T^{4} - 44168 T^{6} + 2167 p^{2} T^{8} - 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 1639 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 2 T^{2} + 979 T^{4} + 164 p T^{6} + 979 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{6} \) |
| 31 | \( ( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3727 p^{2} T^{8} - 58 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 2 T + 47 T^{2} - 276 T^{3} + 47 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 132 T^{2} + 10823 T^{4} - 12872 p T^{6} + 10823 p^{2} T^{8} - 132 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 4567 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 4591 p^{2} T^{8} + 10 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 16171 p^{2} T^{8} - 184 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 7799 p^{2} T^{8} + 146 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 14 T + 187 T^{2} + 1700 T^{3} + 187 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 13175 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 5683 p^{2} T^{8} + 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 47 T^{2} + 352 T^{3} + 47 p T^{4} + p^{3} T^{6} )^{4} \) |
| 79 | \( ( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 19359 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 65191 p^{2} T^{8} + 370 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 49895 p^{2} T^{8} - 308 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 + 135 T^{2} - 128 T^{3} + 135 p T^{4} + p^{3} T^{6} )^{4} \) |
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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
−2.81627273929989317936858687537, −2.79548133572361516618870394425, −2.73515321378192857812259653857, −2.65599623532745281161919188110, −2.56436412334731509751620966437, −2.54385274336286086444442798381, −2.52471357162778011318777941178, −2.41407879704446431492182406234, −2.20141982132110899645995472719, −1.91950071872743811978946926981, −1.83119957283449078386139041884, −1.76828319899767988496799966651, −1.75289475201503552706015303666, −1.66820493613374074843700494452, −1.46191854332025574474046500738, −1.42699217580657860963683528900, −1.39778406206600076957329166001, −1.16047868316533492007008091503, −1.14991667179398320719532754293, −1.12427025089337348054896904737, −0.897890499083998098212942143357, −0.47499905773494448677431459237, −0.41557429960314004329477087881, −0.38228535907633977252529605905, −0.25084258628860063816319786474,
0.25084258628860063816319786474, 0.38228535907633977252529605905, 0.41557429960314004329477087881, 0.47499905773494448677431459237, 0.897890499083998098212942143357, 1.12427025089337348054896904737, 1.14991667179398320719532754293, 1.16047868316533492007008091503, 1.39778406206600076957329166001, 1.42699217580657860963683528900, 1.46191854332025574474046500738, 1.66820493613374074843700494452, 1.75289475201503552706015303666, 1.76828319899767988496799966651, 1.83119957283449078386139041884, 1.91950071872743811978946926981, 2.20141982132110899645995472719, 2.41407879704446431492182406234, 2.52471357162778011318777941178, 2.54385274336286086444442798381, 2.56436412334731509751620966437, 2.65599623532745281161919188110, 2.73515321378192857812259653857, 2.79548133572361516618870394425, 2.81627273929989317936858687537
Plot not available for L-functions of degree greater than 10.
