| L(s) = 1 | − 2·4-s − 2·9-s + 6·11-s + 3·16-s − 2·19-s − 12·29-s + 2·31-s + 4·36-s + 6·41-s − 12·44-s + 11·49-s − 6·59-s + 2·61-s − 4·64-s + 6·71-s + 4·76-s + 10·79-s + 3·81-s − 12·89-s − 12·99-s + 18·101-s − 2·109-s + 24·116-s − 5·121-s − 4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s + 1.80·11-s + 3/4·16-s − 0.458·19-s − 2.22·29-s + 0.359·31-s + 2/3·36-s + 0.937·41-s − 1.80·44-s + 11/7·49-s − 0.781·59-s + 0.256·61-s − 1/2·64-s + 0.712·71-s + 0.458·76-s + 1.12·79-s + 1/3·81-s − 1.27·89-s − 1.20·99-s + 1.79·101-s − 0.191·109-s + 2.22·116-s − 0.454·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095398454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095398454\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 11 T^{2} + 120 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 71 T^{2} + 2244 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 71 T^{2} + 3564 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 131 T^{2} + 8040 T^{4} - 131 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 239 T^{2} + 23052 T^{4} - 239 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 311 T^{2} + 37884 T^{4} - 311 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 6 T + 154 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 24894 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.48988476649999932679678624958, −5.89139532141344706614779335656, −5.88366384354115777490363533050, −5.77068879690959970073818222810, −5.61248554198752612652071989732, −5.25888771762156118942773963127, −4.91236418282782427833711968368, −4.89936652701205194463596356839, −4.81637470674570990891967353079, −4.15586574744672136747092545990, −4.09731127560463502513090494163, −4.04914355762185485333017565869, −3.98471026516833253346243122353, −3.52450324864135045525625314546, −3.35432309237983197299524669203, −3.20336107258129022956583148183, −2.82586232763729123950551301710, −2.37645657570172212763872286168, −2.36711450068850214459224623059, −2.03420409575606134297047951960, −1.63427384201488568092796732470, −1.29503980217968573601059960113, −1.15170760478823635423787717326, −0.58728185645740419556385763219, −0.32445121078107987619924142662,
0.32445121078107987619924142662, 0.58728185645740419556385763219, 1.15170760478823635423787717326, 1.29503980217968573601059960113, 1.63427384201488568092796732470, 2.03420409575606134297047951960, 2.36711450068850214459224623059, 2.37645657570172212763872286168, 2.82586232763729123950551301710, 3.20336107258129022956583148183, 3.35432309237983197299524669203, 3.52450324864135045525625314546, 3.98471026516833253346243122353, 4.04914355762185485333017565869, 4.09731127560463502513090494163, 4.15586574744672136747092545990, 4.81637470674570990891967353079, 4.89936652701205194463596356839, 4.91236418282782427833711968368, 5.25888771762156118942773963127, 5.61248554198752612652071989732, 5.77068879690959970073818222810, 5.88366384354115777490363533050, 5.89139532141344706614779335656, 6.48988476649999932679678624958
