L(s) = 1 | + 2-s − 4-s − 4·5-s + 8·7-s − 3·8-s + 9-s − 4·10-s + 11-s + 8·14-s − 16-s + 18-s + 4·20-s + 22-s + 2·25-s − 8·28-s + 5·32-s − 32·35-s − 36-s + 12·37-s + 12·40-s − 44-s − 4·45-s + 34·49-s + 2·50-s + 12·53-s − 4·55-s − 24·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.78·5-s + 3.02·7-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 2.13·14-s − 1/4·16-s + 0.235·18-s + 0.894·20-s + 0.213·22-s + 2/5·25-s − 1.51·28-s + 0.883·32-s − 5.40·35-s − 1/6·36-s + 1.97·37-s + 1.89·40-s − 0.150·44-s − 0.596·45-s + 34/7·49-s + 0.282·50-s + 1.64·53-s − 0.539·55-s − 3.20·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.020786204\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.020786204\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.975261543279013968212899152873, −8.388317550066773340937260013490, −8.021755838874073182736536348782, −7.890643100699516116601167538593, −7.46995816184461358823952634116, −6.80474099018451965837235050932, −5.99052260380604420685058602068, −5.31622482563829752699303340190, −5.03520265044937013447420220349, −4.30674031395560359136174367267, −4.20407903283329927801590362766, −3.84602405282169040782497156344, −2.78748407328752201898486204249, −1.85717409091206548414350568811, −0.889704343842193223159753487670,
0.889704343842193223159753487670, 1.85717409091206548414350568811, 2.78748407328752201898486204249, 3.84602405282169040782497156344, 4.20407903283329927801590362766, 4.30674031395560359136174367267, 5.03520265044937013447420220349, 5.31622482563829752699303340190, 5.99052260380604420685058602068, 6.80474099018451965837235050932, 7.46995816184461358823952634116, 7.890643100699516116601167538593, 8.021755838874073182736536348782, 8.388317550066773340937260013490, 8.975261543279013968212899152873