| 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
| \(L(1)\) |
\(\approx\) |
\(15461.04676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15461.04676\) |
| \(L(\frac{3}{2})\) |
|
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
−3.26159944611593550510067415173, −3.19404456357144715520940100948, −3.10632659476775170486891240706, −3.04613986804789430790926711197, −2.91886960029965921531338902539, −2.91535021999006765735239057645, −2.67125908763428994749090689218, −2.65826475882876974152882287679, −2.58193790709379258734195970460, −2.44265837455016175337920805511, −2.35466556014619289450650766168, −2.27481677897418109373972890787, −2.17574730889154037426438247923, −1.99071106830388782994112523682, −1.95285649869060385836475227809, −1.79706509620442974308830948049, −1.65655707816319752522146437262, −1.50999325530287141922778632400, −1.49730318305411164768861152720, −1.39945618027725773384606748354, −1.09516167104838738294961158191, −0.970764860927149882269891879603, −0.946101667346293991653068634513, −0.65390603369604191812260471002, −0.42652629392203022099145822970,
0.42652629392203022099145822970, 0.65390603369604191812260471002, 0.946101667346293991653068634513, 0.970764860927149882269891879603, 1.09516167104838738294961158191, 1.39945618027725773384606748354, 1.49730318305411164768861152720, 1.50999325530287141922778632400, 1.65655707816319752522146437262, 1.79706509620442974308830948049, 1.95285649869060385836475227809, 1.99071106830388782994112523682, 2.17574730889154037426438247923, 2.27481677897418109373972890787, 2.35466556014619289450650766168, 2.44265837455016175337920805511, 2.58193790709379258734195970460, 2.65826475882876974152882287679, 2.67125908763428994749090689218, 2.91535021999006765735239057645, 2.91886960029965921531338902539, 3.04613986804789430790926711197, 3.10632659476775170486891240706, 3.19404456357144715520940100948, 3.26159944611593550510067415173
Plot not available for L-functions of degree greater than 10.
