| 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 − 0.206·29-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8161460708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8161460708\) |
| \(L(2)\) |
|
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
−7.66919616807909272490010407077, −7.56815286559497715834819810098, −7.40711610405656000475923044311, −7.32140899650579445272995559863, −7.09942845457914720134534150038, −6.81935945301705152760363547873, −6.28665239891537787514119482973, −5.95708051853623783458306795656, −5.80077040297752502713329763876, −5.50106251309651368607549063664, −5.28962584591007486068151929765, −4.87199954324235508398743635784, −4.83341134545110310447171558243, −4.62356614784607801651047292307, −4.11230681708829124920011807807, −3.83956351015071262929312057889, −3.55062361428722975594833719076, −3.45592004482394501947777154536, −2.68383763551765627182641091925, −2.65930282161934989832252513834, −2.15243820200788307346026128428, −2.01786575154539002977980356957, −1.33301649413595707643002438697, −0.70578592482523940787914997541, −0.24118316578607707151304080353,
0.24118316578607707151304080353, 0.70578592482523940787914997541, 1.33301649413595707643002438697, 2.01786575154539002977980356957, 2.15243820200788307346026128428, 2.65930282161934989832252513834, 2.68383763551765627182641091925, 3.45592004482394501947777154536, 3.55062361428722975594833719076, 3.83956351015071262929312057889, 4.11230681708829124920011807807, 4.62356614784607801651047292307, 4.83341134545110310447171558243, 4.87199954324235508398743635784, 5.28962584591007486068151929765, 5.50106251309651368607549063664, 5.80077040297752502713329763876, 5.95708051853623783458306795656, 6.28665239891537787514119482973, 6.81935945301705152760363547873, 7.09942845457914720134534150038, 7.32140899650579445272995559863, 7.40711610405656000475923044311, 7.56815286559497715834819810098, 7.66919616807909272490010407077
