L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 8·15-s + 4·17-s + 8·19-s − 4·23-s + 4·25-s + 4·27-s + 16·31-s − 8·33-s − 8·37-s + 4·41-s + 16·43-s − 12·45-s + 8·47-s + 8·51-s − 4·53-s + 16·55-s + 16·57-s + 16·59-s − 8·61-s + 16·67-s − 8·69-s − 4·71-s + 8·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s + 0.769·27-s + 2.87·31-s − 1.39·33-s − 1.31·37-s + 0.624·41-s + 2.43·43-s − 1.78·45-s + 1.16·47-s + 1.12·51-s − 0.549·53-s + 2.15·55-s + 2.11·57-s + 2.08·59-s − 1.02·61-s + 1.95·67-s − 0.963·69-s − 0.474·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.047588157\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.047588157\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 118 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 84 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.051833350969874694704352360075, −8.708158176051700115869362542776, −8.117807387412068599463137070272, −8.021337914721385923227782978657, −7.75490448730835657912510225579, −7.52597829640397611346636251485, −7.14775031270455405022195628679, −6.65742998796066199752008478439, −5.98732119808227100832063966567, −5.67248094492287799687207376224, −4.95533941277000983094691603980, −4.86772632215215173156181422525, −4.05938233261773429115816266020, −3.96049629492318870374659752523, −3.27871121143100104399319258404, −3.22841826178006948893098390296, −2.41649035239950971977131476626, −2.25309203672834778486684851137, −0.975969310720986727077965539905, −0.71359646878133320903381307532,
0.71359646878133320903381307532, 0.975969310720986727077965539905, 2.25309203672834778486684851137, 2.41649035239950971977131476626, 3.22841826178006948893098390296, 3.27871121143100104399319258404, 3.96049629492318870374659752523, 4.05938233261773429115816266020, 4.86772632215215173156181422525, 4.95533941277000983094691603980, 5.67248094492287799687207376224, 5.98732119808227100832063966567, 6.65742998796066199752008478439, 7.14775031270455405022195628679, 7.52597829640397611346636251485, 7.75490448730835657912510225579, 8.021337914721385923227782978657, 8.117807387412068599463137070272, 8.708158176051700115869362542776, 9.051833350969874694704352360075