| L(s) = 1 | − 2·2-s + 4-s − 4·7-s − 2·8-s − 3·9-s + 8·14-s + 7·16-s − 12·17-s + 6·18-s + 8·23-s − 4·28-s − 12·31-s − 10·32-s + 24·34-s − 3·36-s − 20·41-s − 16·46-s − 8·47-s + 2·49-s + 8·56-s + 24·62-s + 12·63-s + 13·64-s − 12·68-s − 8·71-s + 6·72-s + 36·73-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.51·7-s − 0.707·8-s − 9-s + 2.13·14-s + 7/4·16-s − 2.91·17-s + 1.41·18-s + 1.66·23-s − 0.755·28-s − 2.15·31-s − 1.76·32-s + 4.11·34-s − 1/2·36-s − 3.12·41-s − 2.35·46-s − 1.16·47-s + 2/7·49-s + 1.06·56-s + 3.04·62-s + 1.51·63-s + 13/8·64-s − 1.45·68-s − 0.949·71-s + 0.707·72-s + 4.21·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1664054174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1664054174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T + 3 T^{2} + 3 p T^{3} + 3 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 2 T + 5 T^{2} + 12 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 11 | \( 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12} \) |
| 13 | \( 1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 407 p^{2} T^{8} - 22 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( ( 1 + 6 T + 35 T^{2} + 172 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 19 | \( 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 - 4 T + 57 T^{2} - 168 T^{3} + 57 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{3} \) |
| 31 | \( ( 1 + 6 T + 77 T^{2} + 308 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 6055 p^{2} T^{8} - 86 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 8087 p^{2} T^{8} - 114 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( ( 1 + 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 53 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{3} \) |
| 59 | \( 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12} \) |
| 67 | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{3} \) |
| 71 | \( ( 1 + 4 T + 101 T^{2} + 632 T^{3} + 101 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( ( 1 - 6 T + p T^{2} )^{6} \) |
| 79 | \( ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 37415 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 263 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( ( 1 + 18 T + 287 T^{2} + 3164 T^{3} + 287 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
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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
−5.99771603995393831709951691866, −5.32915562520870735439483302970, −5.26757605239884486087795153915, −5.21696762738697426075711321086, −5.21351481611910029991471441419, −5.09491099169146447756957214220, −4.80050464605182267642646143433, −4.44402680576877082698655176151, −4.12792965582564463549581956595, −4.11422011800851363223938100233, −3.95118443388469199361381746126, −3.49924428836250819257477577853, −3.48783034461214496032036352762, −3.41806769612818077528683422137, −3.00927015937391533216004507006, −2.99840196858754915576763739664, −2.70964227011762655971627405324, −2.54372439705391079056865300081, −2.08732678671105850208355199259, −2.02236488104300770200098635607, −1.69092314152086572037839665490, −1.55453057656037348169259121235, −0.873423304839546583199217713719, −0.40122481678522826397575925102, −0.25918385832446682173167772046,
0.25918385832446682173167772046, 0.40122481678522826397575925102, 0.873423304839546583199217713719, 1.55453057656037348169259121235, 1.69092314152086572037839665490, 2.02236488104300770200098635607, 2.08732678671105850208355199259, 2.54372439705391079056865300081, 2.70964227011762655971627405324, 2.99840196858754915576763739664, 3.00927015937391533216004507006, 3.41806769612818077528683422137, 3.48783034461214496032036352762, 3.49924428836250819257477577853, 3.95118443388469199361381746126, 4.11422011800851363223938100233, 4.12792965582564463549581956595, 4.44402680576877082698655176151, 4.80050464605182267642646143433, 5.09491099169146447756957214220, 5.21351481611910029991471441419, 5.21696762738697426075711321086, 5.26757605239884486087795153915, 5.32915562520870735439483302970, 5.99771603995393831709951691866
Plot not available for L-functions of degree greater than 10.
