| L(s) = 1 | − 3·5-s + 12·17-s + 12·19-s + 30·23-s − 16·31-s − 48·47-s − 137·49-s − 384·53-s − 38·61-s + 6·79-s + 288·83-s − 36·85-s − 36·95-s + 36·107-s − 452·109-s + 564·113-s − 90·115-s − 129·121-s + 177·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + ⋯ |
| L(s) = 1 | − 3/5·5-s + 0.705·17-s + 0.631·19-s + 1.30·23-s − 0.516·31-s − 1.02·47-s − 2.79·49-s − 7.24·53-s − 0.622·61-s + 6/79·79-s + 3.46·83-s − 0.423·85-s − 0.378·95-s + 0.336·107-s − 4.14·109-s + 4.99·113-s − 0.782·115-s − 1.06·121-s + 1.41·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.309·155-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2045900202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2045900202\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 3 T + 9 T^{2} - 6 p^{2} T^{3} - 34 p^{2} T^{4} - 6 p^{4} T^{5} + 9 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 7 | \( 1 + 137 T^{2} + 9745 T^{4} + 578414 T^{6} + 30603406 T^{8} + 578414 p^{4} T^{10} + 9745 p^{8} T^{12} + 137 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( 1 + 129 T^{2} + 6241 T^{4} - 2435778 T^{6} - 349839762 T^{8} - 2435778 p^{4} T^{10} + 6241 p^{8} T^{12} + 129 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 + 136 T^{2} + 1882 p T^{4} - 8580512 T^{6} - 1308354077 T^{8} - 8580512 p^{4} T^{10} + 1882 p^{9} T^{12} + 136 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 - 3 T + 528 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 3 T + 254 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 - 15 T - 837 T^{2} - 60 T^{3} + 728978 T^{4} - 60 p^{2} T^{5} - 837 p^{4} T^{6} - 15 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 98 T^{2} - 697677 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T + 7 T^{2} - 14920 T^{3} - 981776 T^{4} - 14920 p^{2} T^{5} + 7 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 1522 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 3186 T^{2} + 7324835 T^{4} + 3186 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( 1 + 776 T^{2} - 5114414 T^{4} - 869905312 T^{6} + 18932481039523 T^{8} - 869905312 p^{4} T^{10} - 5114414 p^{8} T^{12} + 776 p^{12} T^{14} + p^{16} T^{16} \) |
| 47 | \( ( 1 + 24 T - 3150 T^{2} - 16608 T^{3} + 7731011 T^{4} - 16608 p^{2} T^{5} - 3150 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 96 T + 6041 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 59 | \( 1 + 7044 T^{2} + 20683306 T^{4} + 33106151952 T^{6} + 89143933045683 T^{8} + 33106151952 p^{4} T^{10} + 20683306 p^{8} T^{12} + 7044 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 19 T - 6701 T^{2} - 7220 T^{3} + 34682722 T^{4} - 7220 p^{2} T^{5} - 6701 p^{4} T^{6} + 19 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 + 6584 T^{2} + 23998450 T^{4} - 137945571424 T^{6} - 905153797681181 T^{8} - 137945571424 p^{4} T^{10} + 23998450 p^{8} T^{12} + 6584 p^{12} T^{14} + p^{16} T^{16} \) |
| 71 | \( ( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 3 T - 8243 T^{2} + 12690 T^{3} + 29089254 T^{4} + 12690 p^{2} T^{5} - 8243 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( ( 1 - 144 T + 6999 T^{2} + 5904 T^{3} - 1603456 T^{4} + 5904 p^{2} T^{5} + 6999 p^{4} T^{6} - 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 - 1879 T^{2} - 50655359 T^{4} + 230877543998 T^{6} - 5213876107576082 T^{8} + 230877543998 p^{4} T^{10} - 50655359 p^{8} T^{12} - 1879 p^{12} T^{14} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.64963194233573135839246710035, −3.56115910347085332928728209966, −3.51319670654781445235677818877, −3.36722113072823609771893214636, −3.28329294938673902991104442801, −3.10991942238344978469552768867, −3.07274129609613386980753755481, −2.98786306064295390816670749962, −2.81697801996612837000380566725, −2.69701529393741040415782136173, −2.32878465329913797451133335443, −2.28757545851302048253133263552, −2.20260162073375897606713405638, −1.95588953408488016418086060901, −1.94654411196773508287557183403, −1.66930903493941138148208592695, −1.39069526861945396801909255515, −1.35536114569284174538826150344, −1.31546278256721780436118786834, −1.19847674694339362791048389883, −0.916592651593512075954171540181, −0.78791244856913012471304110850, −0.32566414524604669195149553344, −0.21960836609003052322858508656, −0.06154048703490492657785609514,
0.06154048703490492657785609514, 0.21960836609003052322858508656, 0.32566414524604669195149553344, 0.78791244856913012471304110850, 0.916592651593512075954171540181, 1.19847674694339362791048389883, 1.31546278256721780436118786834, 1.35536114569284174538826150344, 1.39069526861945396801909255515, 1.66930903493941138148208592695, 1.94654411196773508287557183403, 1.95588953408488016418086060901, 2.20260162073375897606713405638, 2.28757545851302048253133263552, 2.32878465329913797451133335443, 2.69701529393741040415782136173, 2.81697801996612837000380566725, 2.98786306064295390816670749962, 3.07274129609613386980753755481, 3.10991942238344978469552768867, 3.28329294938673902991104442801, 3.36722113072823609771893214636, 3.51319670654781445235677818877, 3.56115910347085332928728209966, 3.64963194233573135839246710035
Plot not available for L-functions of degree greater than 10.
