| L(s) = 1 | − 4-s − 6·5-s + 4·7-s + 9-s − 2·13-s − 3·16-s + 6·20-s + 12·23-s + 17·25-s − 4·28-s − 6·29-s − 24·35-s − 36-s − 6·45-s − 2·49-s + 2·52-s − 18·53-s + 12·59-s + 4·63-s + 7·64-s + 12·65-s + 16·67-s + 18·80-s − 8·81-s − 12·83-s − 8·91-s − 12·92-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.68·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s − 3/4·16-s + 1.34·20-s + 2.50·23-s + 17/5·25-s − 0.755·28-s − 1.11·29-s − 4.05·35-s − 1/6·36-s − 0.894·45-s − 2/7·49-s + 0.277·52-s − 2.47·53-s + 1.56·59-s + 0.503·63-s + 7/8·64-s + 1.48·65-s + 1.95·67-s + 2.01·80-s − 8/9·81-s − 1.31·83-s − 0.838·91-s − 1.25·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3797422817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3797422817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 29 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−17.39588301424158933875677218648, −16.99778015087597905058112764402, −16.06488766482125120963498063711, −15.64798762633737740755367231951, −15.06413860942384192522890506409, −14.71890083440623415685845527252, −14.07619189428567131477170846308, −12.86797184750678925981940018018, −12.63899564306666386775870220650, −11.47438908687119495406679323194, −11.36178678057320903112498398069, −11.05371410469819666640288849456, −9.643088170761776087576267011655, −8.662965289571010527502610511555, −8.215421025489560642376724408658, −7.45878499055687683322781099288, −7.07243239456893174212750743030, −4.92263227468655885794947561274, −4.60354096694864069732607307798, −3.55916987895440262001177104646,
3.55916987895440262001177104646, 4.60354096694864069732607307798, 4.92263227468655885794947561274, 7.07243239456893174212750743030, 7.45878499055687683322781099288, 8.215421025489560642376724408658, 8.662965289571010527502610511555, 9.643088170761776087576267011655, 11.05371410469819666640288849456, 11.36178678057320903112498398069, 11.47438908687119495406679323194, 12.63899564306666386775870220650, 12.86797184750678925981940018018, 14.07619189428567131477170846308, 14.71890083440623415685845527252, 15.06413860942384192522890506409, 15.64798762633737740755367231951, 16.06488766482125120963498063711, 16.99778015087597905058112764402, 17.39588301424158933875677218648
