| L(s) = 1 | − 6·3-s + 3·4-s + 19·9-s − 18·12-s + 3·16-s + 10·17-s + 20·25-s − 46·27-s + 20·29-s + 57·36-s + 22·43-s − 18·48-s − 18·49-s − 60·51-s + 8·61-s − 2·64-s + 30·68-s − 120·75-s − 72·79-s + 121·81-s − 120·87-s + 60·100-s + 12·101-s − 120·103-s − 2·107-s − 138·108-s − 6·113-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 3/2·4-s + 19/3·9-s − 5.19·12-s + 3/4·16-s + 2.42·17-s + 4·25-s − 8.85·27-s + 3.71·29-s + 19/2·36-s + 3.35·43-s − 2.59·48-s − 2.57·49-s − 8.40·51-s + 1.02·61-s − 1/4·64-s + 3.63·68-s − 13.8·75-s − 8.10·79-s + 13.4·81-s − 12.8·87-s + 6·100-s + 1.19·101-s − 11.8·103-s − 0.193·107-s − 13.2·108-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.433782752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.433782752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 + p T + 4 T^{2} + 5 T^{3} - 5 T^{4} - 38 T^{5} - 77 T^{6} - 38 p T^{7} - 5 p^{2} T^{8} + 5 p^{3} T^{9} + 4 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( ( 1 - 2 p T^{2} + 71 T^{4} - 396 T^{6} + 71 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 7 | \( 1 + 18 T^{2} + 181 T^{4} + 122 p T^{6} - 2038 T^{8} - 86054 T^{10} - 805027 T^{12} - 86054 p^{2} T^{14} - 2038 p^{4} T^{16} + 122 p^{7} T^{18} + 181 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24} \) |
| 11 | \( 1 + 49 T^{2} + 1240 T^{4} + 24535 T^{6} + 414767 T^{8} + 5731768 T^{10} + 66986969 T^{12} + 5731768 p^{2} T^{14} + 414767 p^{4} T^{16} + 24535 p^{6} T^{18} + 1240 p^{8} T^{20} + 49 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 - 5 T - 4 T^{2} + T^{3} - p T^{4} + 1368 T^{5} - 6367 T^{6} + 1368 p T^{7} - p^{3} T^{8} + p^{3} T^{9} - 4 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 19 | \( 1 + 81 T^{2} + 3424 T^{4} + 105551 T^{6} + 2692199 T^{8} + 60092368 T^{10} + 1203016601 T^{12} + 60092368 p^{2} T^{14} + 2692199 p^{4} T^{16} + 105551 p^{6} T^{18} + 3424 p^{8} T^{20} + 81 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( ( 1 - 41 T^{2} - 112 T^{3} + 738 T^{4} + 2296 T^{5} - 11193 T^{6} + 2296 p T^{7} + 738 p^{2} T^{8} - 112 p^{3} T^{9} - 41 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 10 T + 17 T^{2} + 122 T^{3} - 278 T^{4} + 222 T^{5} - 8527 T^{6} + 222 p T^{7} - 278 p^{2} T^{8} + 122 p^{3} T^{9} + 17 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 82 T^{2} + 3967 T^{4} - 137116 T^{6} + 3967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( 1 + 138 T^{2} + 9373 T^{4} + 462638 T^{6} + 20123402 T^{8} + 820867474 T^{10} + 31478329205 T^{12} + 820867474 p^{2} T^{14} + 20123402 p^{4} T^{16} + 462638 p^{6} T^{18} + 9373 p^{8} T^{20} + 138 p^{10} T^{22} + p^{12} T^{24} \) |
| 41 | \( 1 + 169 T^{2} + 14504 T^{4} + 947519 T^{6} + 53293799 T^{8} + 2528174712 T^{10} + 106506710241 T^{12} + 2528174712 p^{2} T^{14} + 53293799 p^{4} T^{16} + 947519 p^{6} T^{18} + 14504 p^{8} T^{20} + 169 p^{10} T^{22} + p^{12} T^{24} \) |
| 43 | \( ( 1 - 11 T - 4 T^{2} + 515 T^{3} - 145 T^{4} - 20214 T^{5} + 156435 T^{6} - 20214 p T^{7} - 145 p^{2} T^{8} + 515 p^{3} T^{9} - 4 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 - 206 T^{2} + 20175 T^{4} - 1193924 T^{6} + 20175 p^{2} T^{8} - 206 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 131 T^{2} - 56 T^{3} + 131 p T^{4} + p^{3} T^{6} )^{4} \) |
| 59 | \( 1 + 225 T^{2} + 26016 T^{4} + 2072943 T^{6} + 130229727 T^{8} + 6801435144 T^{10} + 362782063577 T^{12} + 6801435144 p^{2} T^{14} + 130229727 p^{4} T^{16} + 2072943 p^{6} T^{18} + 26016 p^{8} T^{20} + 225 p^{10} T^{22} + p^{12} T^{24} \) |
| 61 | \( ( 1 - 4 T - 23 T^{2} + 692 T^{3} - 2554 T^{4} - 12388 T^{5} + 481421 T^{6} - 12388 p T^{7} - 2554 p^{2} T^{8} + 692 p^{3} T^{9} - 23 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 67 | \( 1 + 129 T^{2} + 3360 T^{4} - 568049 T^{6} - 30377169 T^{8} + 2366904456 T^{10} + 354870011049 T^{12} + 2366904456 p^{2} T^{14} - 30377169 p^{4} T^{16} - 568049 p^{6} T^{18} + 3360 p^{8} T^{20} + 129 p^{10} T^{22} + p^{12} T^{24} \) |
| 71 | \( 1 + 322 T^{2} + 57397 T^{4} + 6820198 T^{6} + 607594826 T^{8} + 45013312330 T^{10} + 3159779132573 T^{12} + 45013312330 p^{2} T^{14} + 607594826 p^{4} T^{16} + 6820198 p^{6} T^{18} + 57397 p^{8} T^{20} + 322 p^{10} T^{22} + p^{12} T^{24} \) |
| 73 | \( ( 1 - 265 T^{2} + 36313 T^{4} - 3192361 T^{6} + 36313 p^{2} T^{8} - 265 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 18 T + 261 T^{2} + 2612 T^{3} + 261 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 - 465 T^{2} + 92609 T^{4} - 10109729 T^{6} + 92609 p^{2} T^{8} - 465 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( 1 + 97 T^{2} - 520 T^{4} - 13169 p T^{6} - 83617537 T^{8} + 1428333264 T^{10} + 762449377521 T^{12} + 1428333264 p^{2} T^{14} - 83617537 p^{4} T^{16} - 13169 p^{7} T^{18} - 520 p^{8} T^{20} + 97 p^{10} T^{22} + p^{12} T^{24} \) |
| 97 | \( 1 + 177 T^{2} + 2872 T^{4} - 1434169 T^{6} - 30639697 T^{8} + 12958954288 T^{10} + 1741641887009 T^{12} + 12958954288 p^{2} T^{14} - 30639697 p^{4} T^{16} - 1434169 p^{6} T^{18} + 2872 p^{8} T^{20} + 177 p^{10} T^{22} + p^{12} T^{24} \) |
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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
−3.96335918201966027538577791188, −3.93638390866002678042739385809, −3.76896283144672556192126185218, −3.72623017369448710439562358245, −3.37439276845529423220860974105, −3.26407539194360817482595225423, −3.13857475819746465688042321092, −2.99180273224166902589167350892, −2.91307919344270683168253199786, −2.91098845561263206184601993181, −2.82487550918472642054633555557, −2.72126438267126118056092333751, −2.69812486875617517268752375351, −2.49244550976883687490772583579, −2.18142688680260506465131609312, −2.11783000863985669687729836949, −1.69760131603564411239270001888, −1.61525478955380364413875893283, −1.56537264957392664277114939279, −1.36392552386706533121100760112, −1.26122412423178952301697580510, −0.992874694009511752353472560510, −0.982213822863960460817008303207, −0.57958388952597065966118921669, −0.42636257839194027975322970398,
0.42636257839194027975322970398, 0.57958388952597065966118921669, 0.982213822863960460817008303207, 0.992874694009511752353472560510, 1.26122412423178952301697580510, 1.36392552386706533121100760112, 1.56537264957392664277114939279, 1.61525478955380364413875893283, 1.69760131603564411239270001888, 2.11783000863985669687729836949, 2.18142688680260506465131609312, 2.49244550976883687490772583579, 2.69812486875617517268752375351, 2.72126438267126118056092333751, 2.82487550918472642054633555557, 2.91098845561263206184601993181, 2.91307919344270683168253199786, 2.99180273224166902589167350892, 3.13857475819746465688042321092, 3.26407539194360817482595225423, 3.37439276845529423220860974105, 3.72623017369448710439562358245, 3.76896283144672556192126185218, 3.93638390866002678042739385809, 3.96335918201966027538577791188
Plot not available for L-functions of degree greater than 10.
