| L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 4·13-s − 4·15-s − 14·19-s + 25-s − 2·27-s − 24·29-s − 2·31-s + 2·37-s + 8·39-s − 4·43-s − 2·45-s − 12·47-s − 28·57-s − 12·59-s − 8·61-s − 8·65-s + 2·67-s + 10·73-s + 2·75-s − 22·79-s − 4·81-s + 24·83-s − 48·87-s − 12·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.10·13-s − 1.03·15-s − 3.21·19-s + 1/5·25-s − 0.384·27-s − 4.45·29-s − 0.359·31-s + 0.328·37-s + 1.28·39-s − 0.609·43-s − 0.298·45-s − 1.75·47-s − 3.70·57-s − 1.56·59-s − 1.02·61-s − 0.992·65-s + 0.244·67-s + 1.17·73-s + 0.230·75-s − 2.47·79-s − 4/9·81-s + 2.63·83-s − 5.14·87-s − 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4536607755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4536607755\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 16 T^{2} - 33 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 28 T^{2} + 255 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T + 13 T^{2} - 142 T^{3} - 1004 T^{4} - 142 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T - 53 T^{2} + 34 T^{3} + 1732 T^{4} + 34 p T^{5} - 53 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 34 T^{2} - 1653 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} + 216 T^{3} + 6519 T^{4} + 216 p T^{5} + 8 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8 T - 2 T^{2} - 448 T^{3} - 3269 T^{4} - 448 p T^{5} - 2 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 34 T^{3} + 8932 T^{4} + 34 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 10 T - 53 T^{2} - 70 T^{3} + 10780 T^{4} - 70 p T^{5} - 53 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T - 52 T^{2} + 216 T^{3} + 17679 T^{4} + 216 p T^{5} - 52 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 186 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.13429693534199913972020394836, −6.13171017425414275100624203647, −5.91620290574726258149948424719, −5.44532724686415631052387095389, −5.30578081348101977113508915615, −5.27152970626604733307767605689, −5.16896728578535197908787825749, −4.51554489987042220441560764059, −4.37908798738001539641590094074, −4.19628370890248774581981754836, −4.16250754471454439850413194818, −3.78543661328838548633756966495, −3.68175796182747367069164149643, −3.58267472778278376187256253464, −3.41120832920248254630264372603, −2.81550433038423142423582757344, −2.70879608646777588186562610033, −2.55936682182247433631083377433, −2.40095626610953786151242087625, −1.70436257213285553929963591493, −1.67583860211877573974846830657, −1.53822671051897831058009885416, −1.44019003848859978796101955880, −0.31597232014307450030315256738, −0.18163567222247178363623773985,
0.18163567222247178363623773985, 0.31597232014307450030315256738, 1.44019003848859978796101955880, 1.53822671051897831058009885416, 1.67583860211877573974846830657, 1.70436257213285553929963591493, 2.40095626610953786151242087625, 2.55936682182247433631083377433, 2.70879608646777588186562610033, 2.81550433038423142423582757344, 3.41120832920248254630264372603, 3.58267472778278376187256253464, 3.68175796182747367069164149643, 3.78543661328838548633756966495, 4.16250754471454439850413194818, 4.19628370890248774581981754836, 4.37908798738001539641590094074, 4.51554489987042220441560764059, 5.16896728578535197908787825749, 5.27152970626604733307767605689, 5.30578081348101977113508915615, 5.44532724686415631052387095389, 5.91620290574726258149948424719, 6.13171017425414275100624203647, 6.13429693534199913972020394836
