| L(s) = 1 | + 4-s + 6·5-s − 7-s + 16-s + 6·20-s + 17·25-s − 28-s − 6·35-s + 4·37-s − 12·41-s − 20·43-s + 12·47-s − 6·49-s + 24·59-s + 64-s + 28·67-s + 16·79-s + 6·80-s − 6·83-s − 36·89-s + 17·100-s − 6·101-s + 4·109-s − 112-s − 13·121-s + 18·125-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.68·5-s − 0.377·7-s + 1/4·16-s + 1.34·20-s + 17/5·25-s − 0.188·28-s − 1.01·35-s + 0.657·37-s − 1.87·41-s − 3.04·43-s + 1.75·47-s − 6/7·49-s + 3.12·59-s + 1/8·64-s + 3.42·67-s + 1.80·79-s + 0.670·80-s − 0.658·83-s − 3.81·89-s + 1.69·100-s − 0.597·101-s + 0.383·109-s − 0.0944·112-s − 1.18·121-s + 1.60·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.900027461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.900027461\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.507801646196580209679416383095, −9.022301181159976542679631550139, −8.268048909391662018895990050852, −8.153752740732516759811873175243, −6.90390849159694968636551446179, −6.73860900674537002760989556093, −6.51972259556560194206176883369, −5.69537931970611061608929469001, −5.37363015537505503550836847886, −5.14293667469015758909535382312, −4.01126047180402972127592124525, −3.26671376711039705407751493003, −2.44850978333263328730546301919, −2.07276435338156744028852711093, −1.33346620954041525748246641205,
1.33346620954041525748246641205, 2.07276435338156744028852711093, 2.44850978333263328730546301919, 3.26671376711039705407751493003, 4.01126047180402972127592124525, 5.14293667469015758909535382312, 5.37363015537505503550836847886, 5.69537931970611061608929469001, 6.51972259556560194206176883369, 6.73860900674537002760989556093, 6.90390849159694968636551446179, 8.153752740732516759811873175243, 8.268048909391662018895990050852, 9.022301181159976542679631550139, 9.507801646196580209679416383095
