| L(s) = 1 | − 6·4-s + 4·7-s + 21·16-s − 24·28-s + 8·37-s − 12·43-s + 22·49-s − 56·64-s + 64·67-s + 56·79-s − 4·81-s − 56·109-s + 84·112-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 48·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 30·169-s + 72·172-s + 173-s + ⋯ |
| L(s) = 1 | − 3·4-s + 1.51·7-s + 21/4·16-s − 4.53·28-s + 1.31·37-s − 1.82·43-s + 22/7·49-s − 7·64-s + 7.81·67-s + 6.30·79-s − 4/9·81-s − 5.36·109-s + 7.93·112-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.94·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.30·169-s + 5.48·172-s + 0.0760·173-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
| \(L(1)\) |
\(\approx\) |
\(9.806842285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.806842285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{6} \) |
| 3 | \( 1 + 4 T^{4} - 10 p T^{6} + 4 p^{2} T^{8} + p^{6} T^{12} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 2 T - 5 T^{2} + 36 T^{3} - 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 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 | \( ( 1 - 73 T^{2} + 105 p T^{4} - 58154 T^{6} + 105 p^{3} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 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 | \( ( 1 - 2 T + p T^{2} - 300 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 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 | \( ( 1 + 3 T + 109 T^{2} + 266 T^{3} + 109 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 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 | \( ( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 13615 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 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 | \( ( 1 - 16 T + 258 T^{2} - 2108 T^{3} + 258 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 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
−3.29533244862797346730239755978, −3.13818200887365177764750535851, −3.01037386089505590694687465923, −2.92660031237026692101455370763, −2.64859415557668962745311956701, −2.54672611404739278976427713901, −2.53965930617689322783954178902, −2.53454687898257615647826247983, −2.43990317470047404311370954475, −2.19051490453833884857933134151, −2.16291686930199594754261515317, −2.10152311085867092828817791321, −1.90954318145687953362884605880, −1.84395676891438595083023054265, −1.65164563765129418582125679595, −1.47385492856451772742696085532, −1.44776973456407214720524802427, −1.27895938125377593109773709556, −1.18810336651684190208766638938, −0.899100208035858701000019826264, −0.74284526345787270661054444972, −0.65629924745685988001081620669, −0.65543857291516831369126118531, −0.39499080563206526711884062032, −0.36552122907014934422579723481,
0.36552122907014934422579723481, 0.39499080563206526711884062032, 0.65543857291516831369126118531, 0.65629924745685988001081620669, 0.74284526345787270661054444972, 0.899100208035858701000019826264, 1.18810336651684190208766638938, 1.27895938125377593109773709556, 1.44776973456407214720524802427, 1.47385492856451772742696085532, 1.65164563765129418582125679595, 1.84395676891438595083023054265, 1.90954318145687953362884605880, 2.10152311085867092828817791321, 2.16291686930199594754261515317, 2.19051490453833884857933134151, 2.43990317470047404311370954475, 2.53454687898257615647826247983, 2.53965930617689322783954178902, 2.54672611404739278976427713901, 2.64859415557668962745311956701, 2.92660031237026692101455370763, 3.01037386089505590694687465923, 3.13818200887365177764750535851, 3.29533244862797346730239755978
Plot not available for L-functions of degree greater than 10.
