| L(s) = 1 | − 3·7-s − 6·13-s + 6·17-s − 3·19-s − 6·23-s + 6·25-s − 24·29-s − 3·31-s − 3·37-s + 12·41-s + 30·43-s + 12·47-s + 9·49-s + 6·53-s + 12·59-s + 18·61-s − 9·67-s − 33·73-s + 27·79-s − 36·83-s − 12·89-s + 18·91-s + 18·101-s − 15·103-s + 24·107-s − 9·109-s − 48·113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1.66·13-s + 1.45·17-s − 0.688·19-s − 1.25·23-s + 6/5·25-s − 4.45·29-s − 0.538·31-s − 0.493·37-s + 1.87·41-s + 4.57·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 1.56·59-s + 2.30·61-s − 1.09·67-s − 3.86·73-s + 3.03·79-s − 3.95·83-s − 1.27·89-s + 1.88·91-s + 1.79·101-s − 1.47·103-s + 2.32·107-s − 0.862·109-s − 4.51·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060513322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060513322\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T - 5 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| good | 5 | \( 1 - 6 T^{2} - 8 T^{3} + 6 T^{4} + 24 T^{5} + 86 T^{6} + 24 p T^{7} + 6 p^{2} T^{8} - 8 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( 1 - 6 T^{2} - 76 T^{3} - 30 T^{4} + 228 T^{5} + 3710 T^{6} + 228 p T^{7} - 30 p^{2} T^{8} - 76 p^{3} T^{9} - 6 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( ( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 17 | \( 1 - 6 T + 9 T^{2} + 54 T^{3} - 378 T^{4} + 858 T^{5} - 1307 T^{6} + 858 p T^{7} - 378 p^{2} T^{8} + 54 p^{3} T^{9} + 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 19 | \( 1 + 3 T - 12 T^{2} + 59 T^{3} + 36 T^{4} - 1269 T^{5} + 2094 T^{6} - 1269 p T^{7} + 36 p^{2} T^{8} + 59 p^{3} T^{9} - 12 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 23 | \( 1 + 6 T - 9 T^{2} - 90 T^{3} - 90 T^{4} - 1146 T^{5} - 8885 T^{6} - 1146 p T^{7} - 90 p^{2} T^{8} - 90 p^{3} T^{9} - 9 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 12 T + 126 T^{2} + 728 T^{3} + 126 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 3 T - 63 T^{2} - 2 p T^{3} + 2535 T^{4} - 501 T^{5} - 90450 T^{6} - 501 p T^{7} + 2535 p^{2} T^{8} - 2 p^{4} T^{9} - 63 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 37 | \( 1 + 3 T - 18 T^{2} + 373 T^{3} + 168 T^{4} - 5829 T^{5} + 83328 T^{6} - 5829 p T^{7} + 168 p^{2} T^{8} + 373 p^{3} T^{9} - 18 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 6 T + 27 T^{2} + 20 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( ( 1 - 15 T + 177 T^{2} - 1318 T^{3} + 177 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 - 12 T - 9 T^{2} + 196 T^{3} + 4026 T^{4} - 9372 T^{5} - 178417 T^{6} - 9372 p T^{7} + 4026 p^{2} T^{8} + 196 p^{3} T^{9} - 9 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 53 | \( 1 - 6 T - 42 T^{2} + 1136 T^{3} - 54 p T^{4} - 29802 T^{5} + 517382 T^{6} - 29802 p T^{7} - 54 p^{3} T^{8} + 1136 p^{3} T^{9} - 42 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} \) |
| 59 | \( 1 - 12 T - 54 T^{2} + 292 T^{3} + 11514 T^{4} - 25536 T^{5} - 623986 T^{6} - 25536 p T^{7} + 11514 p^{2} T^{8} + 292 p^{3} T^{9} - 54 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 61 | \( 1 - 18 T + 129 T^{2} - 222 T^{3} - 3462 T^{4} + 40662 T^{5} - 368431 T^{6} + 40662 p T^{7} - 3462 p^{2} T^{8} - 222 p^{3} T^{9} + 129 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \) |
| 67 | \( 1 + 9 T - 108 T^{2} - 383 T^{3} + 13680 T^{4} + 9405 T^{5} - 1069626 T^{6} + 9405 p T^{7} + 13680 p^{2} T^{8} - 383 p^{3} T^{9} - 108 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 177 T^{2} + 32 T^{3} + 177 p T^{4} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 + 33 T + 534 T^{2} + 6687 T^{3} + 75648 T^{4} + 728637 T^{5} + 6291524 T^{6} + 728637 p T^{7} + 75648 p^{2} T^{8} + 6687 p^{3} T^{9} + 534 p^{4} T^{10} + 33 p^{5} T^{11} + p^{6} T^{12} \) |
| 79 | \( 1 - 27 T + 297 T^{2} - 2426 T^{3} + 27783 T^{4} - 321003 T^{5} + 3028254 T^{6} - 321003 p T^{7} + 27783 p^{2} T^{8} - 2426 p^{3} T^{9} + 297 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} \) |
| 83 | \( ( 1 + 18 T + 234 T^{2} + 2040 T^{3} + 234 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 89 | \( 1 + 12 T - 135 T^{2} - 700 T^{3} + 28722 T^{4} + 63804 T^{5} - 2561959 T^{6} + 63804 p T^{7} + 28722 p^{2} T^{8} - 700 p^{3} T^{9} - 135 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \) |
| 97 | \( ( 1 + 264 T^{2} + 38 T^{3} + 264 p T^{4} + p^{3} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.73358691473855304666908508792, −4.54041107958358795781235411462, −4.45915024596663105938641234712, −4.27176056514759985365322239329, −4.07186680859602725889803462971, −3.98798079304625088676741011186, −3.97725086844960362205797236678, −3.66112839052579034048417634964, −3.51274598499721509137705356276, −3.45789982026229157690074234889, −3.28532768333680895484636049297, −3.06163463831914158157389095488, −2.59268135541074536305180937797, −2.55988149442604569664450496440, −2.53533863204684771207572698904, −2.53315710657257944682558990194, −2.11486516122628116274736297943, −2.10201233333231931924572374576, −1.90786065007218142949134271486, −1.47929899361279732280347728617, −1.15132733332247512528282218801, −1.01035654160233090890487387945, −0.968880721461103888182901828628, −0.29325182691618461834602913760, −0.23250262492086294853466131560,
0.23250262492086294853466131560, 0.29325182691618461834602913760, 0.968880721461103888182901828628, 1.01035654160233090890487387945, 1.15132733332247512528282218801, 1.47929899361279732280347728617, 1.90786065007218142949134271486, 2.10201233333231931924572374576, 2.11486516122628116274736297943, 2.53315710657257944682558990194, 2.53533863204684771207572698904, 2.55988149442604569664450496440, 2.59268135541074536305180937797, 3.06163463831914158157389095488, 3.28532768333680895484636049297, 3.45789982026229157690074234889, 3.51274598499721509137705356276, 3.66112839052579034048417634964, 3.97725086844960362205797236678, 3.98798079304625088676741011186, 4.07186680859602725889803462971, 4.27176056514759985365322239329, 4.45915024596663105938641234712, 4.54041107958358795781235411462, 4.73358691473855304666908508792
Plot not available for L-functions of degree greater than 10.
