| L(s) = 1 | − 24·13-s − 12·25-s − 6·49-s + 72·61-s − 72·113-s − 60·121-s + 48·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 156·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 6.65·13-s − 2.39·25-s − 6/7·49-s + 9.21·61-s − 6.77·113-s − 5.45·121-s + 4.29·125-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 + 12·169-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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.001437692805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001437692805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 6 T^{2} - 33 T^{4} - 556 T^{6} - 33 p^{2} T^{8} + 6 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 5 | \( ( 1 + 3 T^{2} - 12 T^{3} + 3 p T^{4} + p^{3} T^{6} )^{4} \) |
| 11 | \( ( 1 + 30 T^{2} + 327 T^{4} + 2548 T^{6} + 327 p^{2} T^{8} + 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{12} \) |
| 17 | \( ( 1 - 30 T^{2} + 831 T^{4} - 12628 T^{6} + 831 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 66 T^{2} + 2343 T^{4} - 54844 T^{6} + 2343 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 78 T^{2} + 3567 T^{4} - 98836 T^{6} + 3567 p^{2} T^{8} - 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 42 T^{2} + 1575 T^{4} - 60076 T^{6} + 1575 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 + 54 T^{2} + 3471 T^{4} + 101684 T^{6} + 3471 p^{2} T^{8} + 54 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 126 T^{2} + 7863 T^{4} - 337412 T^{6} + 7863 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 126 T^{2} + 6639 T^{4} - 258580 T^{6} + 6639 p^{2} T^{8} - 126 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 78 T^{2} + 4503 T^{4} + 248420 T^{6} + 4503 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 90 T^{2} + 7791 T^{4} + 391852 T^{6} + 7791 p^{2} T^{8} + 90 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 186 T^{2} + 19575 T^{4} - 1258444 T^{6} + 19575 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 - 258 T^{2} + 31863 T^{4} - 2365180 T^{6} + 31863 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 18 T + 195 T^{2} - 1484 T^{3} + 195 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 + 78 T^{2} + 3399 T^{4} + 64676 T^{6} + 3399 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 71 | \( ( 1 - 222 T^{2} + 23727 T^{4} - 1839796 T^{6} + 23727 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 150 T^{2} + 16575 T^{4} - 1350452 T^{6} + 16575 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 234 T^{2} + 22767 T^{4} - 1649932 T^{6} + 22767 p^{2} T^{8} - 234 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 258 T^{2} + 42087 T^{4} - 4123708 T^{6} + 42087 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 222 T^{2} + 32463 T^{4} - 3429460 T^{6} + 32463 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 390 T^{2} + 77391 T^{4} - 9349652 T^{6} + 77391 p^{2} T^{8} - 390 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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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
−2.54342446375947012770802636310, −2.49479402271405981982089005889, −2.43421425898076652932233304201, −2.35642398642689717788434069491, −2.29981738220395457107883664282, −2.05482207177584761681169477602, −2.02155049578552834038232382972, −1.97271553394563940861736909295, −1.91279954344142854854824916924, −1.81393134748001557270330283478, −1.78437354174787926943007068905, −1.77209272421036550162719582051, −1.55367779793375966101535866063, −1.29410827722097238949445818143, −1.26224546481457221163748071518, −1.16701240149630650568735317085, −1.07226342876867878108114497089, −0.985269156130291950341533152830, −0.75161811497010203026987075404, −0.70212148703502468712327813035, −0.63579899761212797814234305073, −0.49997052174288443335197595928, −0.14422389759729768822488020222, −0.05294677991150152757921431471, −0.02414805292597817478619217031,
0.02414805292597817478619217031, 0.05294677991150152757921431471, 0.14422389759729768822488020222, 0.49997052174288443335197595928, 0.63579899761212797814234305073, 0.70212148703502468712327813035, 0.75161811497010203026987075404, 0.985269156130291950341533152830, 1.07226342876867878108114497089, 1.16701240149630650568735317085, 1.26224546481457221163748071518, 1.29410827722097238949445818143, 1.55367779793375966101535866063, 1.77209272421036550162719582051, 1.78437354174787926943007068905, 1.81393134748001557270330283478, 1.91279954344142854854824916924, 1.97271553394563940861736909295, 2.02155049578552834038232382972, 2.05482207177584761681169477602, 2.29981738220395457107883664282, 2.35642398642689717788434069491, 2.43421425898076652932233304201, 2.49479402271405981982089005889, 2.54342446375947012770802636310
Plot not available for L-functions of degree greater than 10.
