| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s − 9-s − 8·14-s − 4·16-s + 4·17-s − 2·18-s + 16·23-s − 25-s − 8·28-s + 20·31-s − 8·32-s + 8·34-s − 2·36-s − 12·41-s + 32·46-s − 16·47-s − 2·49-s − 2·50-s + 40·62-s + 4·63-s − 8·64-s + 8·68-s − 12·73-s + 28·79-s + 81-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s − 1/3·9-s − 2.13·14-s − 16-s + 0.970·17-s − 0.471·18-s + 3.33·23-s − 1/5·25-s − 1.51·28-s + 3.59·31-s − 1.41·32-s + 1.37·34-s − 1/3·36-s − 1.87·41-s + 4.71·46-s − 2.33·47-s − 2/7·49-s − 0.282·50-s + 5.08·62-s + 0.503·63-s − 64-s + 0.970·68-s − 1.40·73-s + 3.15·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.470751743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.470751743\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.773917693271901883731726422599, −9.400231574224530348384608205510, −9.289624976455758475507280984761, −8.460322663024012764249744979418, −8.308632048710853227545437096851, −7.82210791212403284430234163518, −6.94392819950621208538935216013, −6.79192703593532463761964500283, −6.50588947154595386434785942407, −6.22904816197585660812660063405, −5.44765817367683363816900428469, −5.26048333904737107372727529876, −4.61323595646084852324191768052, −4.57107347894644634128540636025, −3.45685457136843648800472497314, −3.33434962941125365209664665929, −2.95418798254998826638044471352, −2.65388954814628595328583468480, −1.50316679062684066098152859230, −0.62671312723919090397864138413,
0.62671312723919090397864138413, 1.50316679062684066098152859230, 2.65388954814628595328583468480, 2.95418798254998826638044471352, 3.33434962941125365209664665929, 3.45685457136843648800472497314, 4.57107347894644634128540636025, 4.61323595646084852324191768052, 5.26048333904737107372727529876, 5.44765817367683363816900428469, 6.22904816197585660812660063405, 6.50588947154595386434785942407, 6.79192703593532463761964500283, 6.94392819950621208538935216013, 7.82210791212403284430234163518, 8.308632048710853227545437096851, 8.460322663024012764249744979418, 9.289624976455758475507280984761, 9.400231574224530348384608205510, 9.773917693271901883731726422599
