| L(s) = 1 | + 2·2-s + 3·4-s + 8·8-s − 2·9-s + 10·13-s + 9·16-s − 4·18-s − 6·19-s + 7·25-s + 20·26-s + 6·32-s − 6·36-s − 12·38-s + 6·43-s − 28·47-s + 28·49-s + 14·50-s + 30·52-s − 16·53-s − 12·59-s + 11·64-s + 16·67-s − 16·72-s − 18·76-s + 3·81-s + 20·83-s + 12·86-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.82·8-s − 2/3·9-s + 2.77·13-s + 9/4·16-s − 0.942·18-s − 1.37·19-s + 7/5·25-s + 3.92·26-s + 1.06·32-s − 36-s − 1.94·38-s + 0.914·43-s − 4.08·47-s + 4·49-s + 1.97·50-s + 4.16·52-s − 2.19·53-s − 1.56·59-s + 11/8·64-s + 1.95·67-s − 1.88·72-s − 2.06·76-s + 1/3·81-s + 2.19·83-s + 1.29·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.53669782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.53669782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $D_4\times C_2$ | \( 1 - 35 T^{2} + 544 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 43 T^{2} + 1176 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 151 T^{2} + 9024 T^{4} - 151 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 12142 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13894 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T^{2} + 6990 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 256 T^{2} + 31870 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.28724249778668748594292780669, −7.04276444674640602650063498448, −6.71817446494673057844096489761, −6.41562075908726301516969512713, −6.23076007081288866423770318278, −6.10849742413931495094907776192, −5.99941058863323363928885223133, −5.98455268958490365299250845248, −5.21004957894356305996809834004, −4.90562790005371530818292158063, −4.90197516509079343255487019402, −4.87875069422487561836358364365, −4.60936521158899159765914983106, −4.05169951256191581691730623027, −3.80191136902803623806435183164, −3.79964468319149247728129549137, −3.52201291322088320789591099123, −3.07996250754678394068736016396, −3.05457188830752067390363115608, −2.36158011000975145827697861911, −2.09519666535584712411791509894, −1.99375384886021617023743166884, −1.54914932149894301646994621896, −1.07610526593356158965479574924, −0.66928701268447182653032471969,
0.66928701268447182653032471969, 1.07610526593356158965479574924, 1.54914932149894301646994621896, 1.99375384886021617023743166884, 2.09519666535584712411791509894, 2.36158011000975145827697861911, 3.05457188830752067390363115608, 3.07996250754678394068736016396, 3.52201291322088320789591099123, 3.79964468319149247728129549137, 3.80191136902803623806435183164, 4.05169951256191581691730623027, 4.60936521158899159765914983106, 4.87875069422487561836358364365, 4.90197516509079343255487019402, 4.90562790005371530818292158063, 5.21004957894356305996809834004, 5.98455268958490365299250845248, 5.99941058863323363928885223133, 6.10849742413931495094907776192, 6.23076007081288866423770318278, 6.41562075908726301516969512713, 6.71817446494673057844096489761, 7.04276444674640602650063498448, 7.28724249778668748594292780669
