| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 9-s + 4·13-s + 5·16-s − 8·17-s − 2·18-s − 6·19-s + 8·26-s + 6·32-s − 16·34-s − 3·36-s − 12·38-s − 18·43-s + 26·47-s + 13·49-s + 12·52-s + 26·53-s − 16·59-s + 7·64-s + 6·67-s − 24·68-s − 4·72-s − 18·76-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1/3·9-s + 1.10·13-s + 5/4·16-s − 1.94·17-s − 0.471·18-s − 1.37·19-s + 1.56·26-s + 1.06·32-s − 2.74·34-s − 1/2·36-s − 1.94·38-s − 2.74·43-s + 3.79·47-s + 13/7·49-s + 1.66·52-s + 3.57·53-s − 2.08·59-s + 7/8·64-s + 0.733·67-s − 2.91·68-s − 0.471·72-s − 2.06·76-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.473449715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.473449715\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.085514229534168623571347140697, −8.507787607529048968514018093964, −8.477969929882081829309232254434, −8.002118998813068293308421509221, −7.26922240518011578066934802029, −6.92012027664786304904501715828, −6.84295722117328091055574887225, −6.32014709620570422109673912370, −5.79098778409271303510493646373, −5.73988783409543467779156255238, −5.17942714377291361067994938241, −4.67644723648061099339003256220, −4.16175739116996434837335547383, −3.89997433158295670569028975311, −3.80558276262290617038565684433, −2.76402658315793566593689150878, −2.60696094820419574261750020886, −2.10391178750548379092350906820, −1.52158236445101351615644976733, −0.58410090781401167734450168491,
0.58410090781401167734450168491, 1.52158236445101351615644976733, 2.10391178750548379092350906820, 2.60696094820419574261750020886, 2.76402658315793566593689150878, 3.80558276262290617038565684433, 3.89997433158295670569028975311, 4.16175739116996434837335547383, 4.67644723648061099339003256220, 5.17942714377291361067994938241, 5.73988783409543467779156255238, 5.79098778409271303510493646373, 6.32014709620570422109673912370, 6.84295722117328091055574887225, 6.92012027664786304904501715828, 7.26922240518011578066934802029, 8.002118998813068293308421509221, 8.477969929882081829309232254434, 8.507787607529048968514018093964, 9.085514229534168623571347140697
