| L(s) = 1 | + 8·5-s + 12·13-s + 4·17-s + 38·25-s + 20·37-s − 4·53-s + 16·61-s + 96·65-s − 12·73-s + 32·85-s − 12·97-s − 72·101-s − 44·113-s − 20·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 3.32·13-s + 0.970·17-s + 38/5·25-s + 3.28·37-s − 0.549·53-s + 2.04·61-s + 11.9·65-s − 1.40·73-s + 3.47·85-s − 1.21·97-s − 7.16·101-s − 4.13·113-s − 1.81·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.90012473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.90012473\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.54823101825916771162653478111, −6.93027329966505251899263308239, −6.81047690892648088793871774872, −6.77492387315936618083378343711, −6.43241023174306288182190377760, −6.30892968533645490676068413645, −5.87663439525393893898199069515, −5.86288057028163082038441799367, −5.81671477881983820569909819136, −5.41394752360444055875605633043, −5.33599772352351055866390974302, −5.03487519846613444273873738735, −4.68808491534124084129503227756, −4.17912950374236370212943521181, −3.94410361057665538280631075859, −3.94264101926466800700322737591, −3.45815656631332532523092023229, −2.83820755997016284325442560711, −2.78831392544147983224895355407, −2.63852224880232550203143512485, −2.27886909060435536651965762010, −1.59861957531367978836704335989, −1.41645507056146449885809412079, −1.25108832525556024564387860561, −1.01045886221699447086587951208,
1.01045886221699447086587951208, 1.25108832525556024564387860561, 1.41645507056146449885809412079, 1.59861957531367978836704335989, 2.27886909060435536651965762010, 2.63852224880232550203143512485, 2.78831392544147983224895355407, 2.83820755997016284325442560711, 3.45815656631332532523092023229, 3.94264101926466800700322737591, 3.94410361057665538280631075859, 4.17912950374236370212943521181, 4.68808491534124084129503227756, 5.03487519846613444273873738735, 5.33599772352351055866390974302, 5.41394752360444055875605633043, 5.81671477881983820569909819136, 5.86288057028163082038441799367, 5.87663439525393893898199069515, 6.30892968533645490676068413645, 6.43241023174306288182190377760, 6.77492387315936618083378343711, 6.81047690892648088793871774872, 6.93027329966505251899263308239, 7.54823101825916771162653478111
