| L(s) = 1 | − 31·5-s − 73·7-s + 162·9-s − 233·11-s + 353·17-s − 722·19-s + 316·23-s + 625·25-s + 2.26e3·35-s + 3.52e3·43-s − 5.02e3·45-s + 1.20e3·47-s + 2.40e3·49-s + 7.22e3·55-s − 3.16e3·61-s − 1.18e4·63-s + 1.00e4·73-s + 1.70e4·77-s + 1.96e4·81-s + 1.13e4·83-s − 1.09e4·85-s + 2.23e4·95-s − 3.77e4·99-s − 1.99e4·101-s − 9.79e3·115-s − 2.57e4·119-s + 1.46e4·121-s + ⋯ |
| L(s) = 1 | − 1.23·5-s − 1.48·7-s + 2·9-s − 1.92·11-s + 1.22·17-s − 2·19-s + 0.597·23-s + 25-s + 1.84·35-s + 1.90·43-s − 2.47·45-s + 0.546·47-s + 49-s + 2.38·55-s − 0.851·61-s − 2.97·63-s + 1.88·73-s + 2.86·77-s + 3·81-s + 1.64·83-s − 1.51·85-s + 2.47·95-s − 3.85·99-s − 1.96·101-s − 0.740·115-s − 1.81·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8426581739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8426581739\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 31 T + 336 T^{2} + 31 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 73 T + 2928 T^{2} + 73 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 233 T + 39648 T^{2} + 233 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 353 T + 41088 T^{2} - 353 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 158 T + p^{4} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 3527 T + 9020928 T^{2} - 3527 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 1207 T - 3422832 T^{2} - 1207 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 5678 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.95240219815696386526109384201, −10.85892984827640419235482496352, −10.32115537757384892687341971643, −10.14013856401201410031513731708, −9.392200407310255401152114940190, −9.158601694087756824725542624961, −8.107200147727656541209355193167, −8.065892195228873796997326352875, −7.29464165242088584373202853556, −7.22064065071941939568119767965, −6.46842371030224318748053220339, −6.03376298507609306227627918336, −5.12493154858281719177787667525, −4.70829232359955676675288697186, −3.86928592816579607662981692405, −3.78434703301554825974946216427, −2.83370906936667219368000080827, −2.30626994982906196938566675583, −1.08735296575340697381193624608, −0.32603285767085400083839343168,
0.32603285767085400083839343168, 1.08735296575340697381193624608, 2.30626994982906196938566675583, 2.83370906936667219368000080827, 3.78434703301554825974946216427, 3.86928592816579607662981692405, 4.70829232359955676675288697186, 5.12493154858281719177787667525, 6.03376298507609306227627918336, 6.46842371030224318748053220339, 7.22064065071941939568119767965, 7.29464165242088584373202853556, 8.065892195228873796997326352875, 8.107200147727656541209355193167, 9.158601694087756824725542624961, 9.392200407310255401152114940190, 10.14013856401201410031513731708, 10.32115537757384892687341971643, 10.85892984827640419235482496352, 10.95240219815696386526109384201
