| L(s) = 1 | + 6·5-s − 6·7-s − 18·11-s + 14·13-s − 8·19-s − 30·23-s + 3·25-s + 6·29-s + 74·31-s − 36·35-s + 120·37-s + 138·41-s − 10·43-s − 174·47-s + 11·49-s − 108·55-s − 18·59-s + 62·61-s + 84·65-s + 22·67-s + 40·73-s + 108·77-s − 86·79-s − 66·83-s − 84·91-s − 48·95-s + 242·97-s + ⋯ |
| L(s) = 1 | + 6/5·5-s − 6/7·7-s − 1.63·11-s + 1.07·13-s − 0.421·19-s − 1.30·23-s + 3/25·25-s + 6/29·29-s + 2.38·31-s − 1.02·35-s + 3.24·37-s + 3.36·41-s − 0.232·43-s − 3.70·47-s + 0.224·49-s − 1.96·55-s − 0.305·59-s + 1.01·61-s + 1.29·65-s + 0.328·67-s + 0.547·73-s + 1.40·77-s − 1.08·79-s − 0.795·83-s − 0.923·91-s − 0.505·95-s + 2.49·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.391303560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391303560\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 6 T + 33 T^{2} - 126 T^{3} + 116 T^{4} - 126 p^{2} T^{5} + 33 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 25 T^{2} - 522 T^{3} - 4044 T^{4} - 522 p^{2} T^{5} + 25 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 18 T + 249 T^{2} + 2538 T^{3} + 18308 T^{4} + 2538 p^{2} T^{5} + 249 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 14 T - 95 T^{2} + 658 T^{3} + 22996 T^{4} + 658 p^{2} T^{5} - 95 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 135302 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 18 p T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 30 T + 1401 T^{2} + 33030 T^{3} + 1091060 T^{4} + 33030 p^{2} T^{5} + 1401 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T + 1409 T^{2} - 8382 T^{3} + 1254420 T^{4} - 8382 p^{2} T^{5} + 1409 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 74 T + 2281 T^{2} - 94202 T^{3} + 4022068 T^{4} - 94202 p^{2} T^{5} + 2281 p^{4} T^{6} - 74 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 60 T + 3254 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 69 T + 3268 T^{2} - 69 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T - 2087 T^{2} - 15110 T^{3} + 1179268 T^{4} - 15110 p^{2} T^{5} - 2087 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 174 T + 16745 T^{2} + 1157622 T^{3} + 61675956 T^{4} + 1157622 p^{2} T^{5} + 16745 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 996 T^{2} - 9136858 T^{4} - 996 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 18 T + 6969 T^{2} + 123498 T^{3} + 35331908 T^{4} + 123498 p^{2} T^{5} + 6969 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 62 T - 4463 T^{2} - 53630 T^{3} + 40856884 T^{4} - 53630 p^{2} T^{5} - 4463 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22 T + 985 T^{2} + 208538 T^{3} - 22072796 T^{4} + 208538 p^{2} T^{5} + 985 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 16452 T^{2} + 117605702 T^{4} - 16452 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 20 T + 7302 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 86 T - 2231 T^{2} - 245530 T^{3} + 7570612 T^{4} - 245530 p^{2} T^{5} - 2231 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 66 T + 9321 T^{2} + 519354 T^{3} + 24465668 T^{4} + 519354 p^{2} T^{5} + 9321 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 25924 T^{2} + 285535302 T^{4} - 25924 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 242 T + 25489 T^{2} - 3450194 T^{3} + 454397668 T^{4} - 3450194 p^{2} T^{5} + 25489 p^{4} T^{6} - 242 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.49390548900001951940319724759, −6.20390789602903905188883976657, −5.94694775782483659683721295436, −5.79689869101160703263901050923, −5.78859510994078166385068493615, −5.54481195044691104813458150796, −5.16171857271938035107179669192, −4.84259040291651436851628870467, −4.82388618778976018918412192627, −4.31513195132048642340747444130, −4.27204452873666910539818769149, −4.19344069048075078939807105703, −3.81717122987854392817187423575, −3.44749989445467143541125607870, −3.03054441190722984673246068919, −2.93799035521976640228784725560, −2.92046249838482300599648958405, −2.31103504013228149963099954614, −2.24443217292774221977887144493, −2.22047351622039785182921293633, −1.62440547603990938748605767666, −1.18231373429416668209257575379, −1.04086658623673939089848364594, −0.62497047715582913213327895865, −0.20772264429018792308871420732,
0.20772264429018792308871420732, 0.62497047715582913213327895865, 1.04086658623673939089848364594, 1.18231373429416668209257575379, 1.62440547603990938748605767666, 2.22047351622039785182921293633, 2.24443217292774221977887144493, 2.31103504013228149963099954614, 2.92046249838482300599648958405, 2.93799035521976640228784725560, 3.03054441190722984673246068919, 3.44749989445467143541125607870, 3.81717122987854392817187423575, 4.19344069048075078939807105703, 4.27204452873666910539818769149, 4.31513195132048642340747444130, 4.82388618778976018918412192627, 4.84259040291651436851628870467, 5.16171857271938035107179669192, 5.54481195044691104813458150796, 5.78859510994078166385068493615, 5.79689869101160703263901050923, 5.94694775782483659683721295436, 6.20390789602903905188883976657, 6.49390548900001951940319724759
