L(s) = 1 | − 4·2-s + 8·4-s − 2·7-s − 8·8-s − 5·9-s + 2·11-s + 8·14-s − 4·16-s + 20·18-s − 8·22-s − 2·23-s − 9·25-s − 16·28-s + 32·32-s − 40·36-s + 6·37-s − 12·43-s + 16·44-s + 8·46-s − 3·49-s + 36·50-s − 12·53-s + 16·56-s + 10·63-s − 64·64-s − 14·67-s − 6·71-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 4·4-s − 0.755·7-s − 2.82·8-s − 5/3·9-s + 0.603·11-s + 2.13·14-s − 16-s + 4.71·18-s − 1.70·22-s − 0.417·23-s − 9/5·25-s − 3.02·28-s + 5.65·32-s − 6.66·36-s + 0.986·37-s − 1.82·43-s + 2.41·44-s + 1.17·46-s − 3/7·49-s + 5.09·50-s − 1.64·53-s + 2.13·56-s + 1.25·63-s − 8·64-s − 1.71·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−11.45125861034521058353374083886, −11.12654606872452134010733513837, −10.03550909718107888433464868208, −10.03359978338253814456351958606, −9.410787129321073286385580378997, −8.678914216536525529374637909269, −8.603539619290756001226038948684, −7.77729309499855305690824525396, −7.38142851192821190173908719125, −6.36261389471308870138602900888, −5.98958560322778025003846316474, −4.56768346065918811492983607197, −3.18179592020545258264561636452, −1.88740391319498359233714309471, 0,
1.88740391319498359233714309471, 3.18179592020545258264561636452, 4.56768346065918811492983607197, 5.98958560322778025003846316474, 6.36261389471308870138602900888, 7.38142851192821190173908719125, 7.77729309499855305690824525396, 8.603539619290756001226038948684, 8.678914216536525529374637909269, 9.410787129321073286385580378997, 10.03359978338253814456351958606, 10.03550909718107888433464868208, 11.12654606872452134010733513837, 11.45125861034521058353374083886