| L(s) = 1 | + 3·5-s − 3·7-s + 6·13-s − 6·19-s − 3·23-s + 6·25-s + 3·29-s − 6·31-s − 9·35-s + 12·37-s − 3·41-s − 6·43-s + 15·47-s + 3·49-s + 6·53-s + 6·59-s + 21·61-s + 18·65-s − 9·67-s + 24·71-s + 24·73-s − 6·79-s − 21·83-s + 9·89-s − 18·91-s − 18·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.13·7-s + 1.66·13-s − 1.37·19-s − 0.625·23-s + 6/5·25-s + 0.557·29-s − 1.07·31-s − 1.52·35-s + 1.97·37-s − 0.468·41-s − 0.914·43-s + 2.18·47-s + 3/7·49-s + 0.824·53-s + 0.781·59-s + 2.68·61-s + 2.23·65-s − 1.09·67-s + 2.84·71-s + 2.80·73-s − 0.675·79-s − 2.30·83-s + 0.953·89-s − 1.88·91-s − 1.84·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.584549838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.584549838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} - T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 9 T^{2} - 36 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 27 T^{2} - 80 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 27 T^{2} - 36 T^{3} + 27 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 224 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 54 T^{2} + 147 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} + 105 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 81 T^{2} + 368 T^{3} + 81 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 75 T^{2} - 452 T^{3} + 75 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 3 T + 42 T^{2} + 327 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 6 T + 117 T^{2} + 440 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 15 T + 180 T^{2} - 1437 T^{3} + 180 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 99 T^{2} - 708 T^{3} + 99 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 117 T^{2} - 780 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 21 T + 246 T^{2} - 2153 T^{3} + 246 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 9 T + 180 T^{2} + 1055 T^{3} + 180 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 24 T + 321 T^{2} - 3084 T^{3} + 321 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 21 T + 378 T^{2} + 3729 T^{3} + 378 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 327 T^{2} - 3068 T^{3} + 327 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.07131639580734821824789036680, −6.57826284492329442690331692066, −6.56417197442524122523823180802, −6.56338172821207000063961331662, −6.07098122687810217194194477052, −6.00900836753811484690333797628, −5.67546884943044508854964369582, −5.38148497264270093230356846033, −5.28724846955964619473785885025, −5.25019358866499207553467641268, −4.40186459740055727056235630143, −4.29678023792044686039793557732, −4.25778858176113385959465244108, −3.85289494418868693312209438737, −3.66021982627962771117278123613, −3.30477595781931801385145299475, −3.00648610633303029079421901000, −2.75622375562489067684266020523, −2.46459311859824755443546711242, −1.91886772583180075172208663367, −1.87784259467026651470043732555, −1.85867600333001634440653383348, −0.882441195678238684263708154305, −0.71812508252192664438413076101, −0.60732554411577488806603589455,
0.60732554411577488806603589455, 0.71812508252192664438413076101, 0.882441195678238684263708154305, 1.85867600333001634440653383348, 1.87784259467026651470043732555, 1.91886772583180075172208663367, 2.46459311859824755443546711242, 2.75622375562489067684266020523, 3.00648610633303029079421901000, 3.30477595781931801385145299475, 3.66021982627962771117278123613, 3.85289494418868693312209438737, 4.25778858176113385959465244108, 4.29678023792044686039793557732, 4.40186459740055727056235630143, 5.25019358866499207553467641268, 5.28724846955964619473785885025, 5.38148497264270093230356846033, 5.67546884943044508854964369582, 6.00900836753811484690333797628, 6.07098122687810217194194477052, 6.56338172821207000063961331662, 6.56417197442524122523823180802, 6.57826284492329442690331692066, 7.07131639580734821824789036680
