| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 4·5-s + 8·6-s − 4·7-s + 2·8-s + 7·9-s + 8·10-s − 4·11-s − 4·12-s − 2·13-s + 8·14-s + 16·15-s − 3·16-s + 2·17-s − 14·18-s − 12·19-s − 4·20-s + 16·21-s + 8·22-s + 8·23-s − 8·24-s + 6·25-s + 4·26-s − 4·27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 3.26·6-s − 1.51·7-s + 0.707·8-s + 7/3·9-s + 2.52·10-s − 1.20·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s + 4.13·15-s − 3/4·16-s + 0.485·17-s − 3.29·18-s − 2.75·19-s − 0.894·20-s + 3.49·21-s + 1.70·22-s + 1.66·23-s − 1.63·24-s + 6/5·25-s + 0.784·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5618 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5618 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - T + p T^{2} ) \) |
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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
−17.8739763542, −16.9988232526, −16.9697718502, −16.5663453795, −16.3347997060, −15.4626793990, −15.3214242661, −14.6517399504, −13.2192960322, −12.8115084431, −12.6214261523, −11.9135244006, −11.1708289929, −11.1205416798, −10.3253836772, −10.1489334004, −9.26423882619, −8.43899013961, −8.00098977891, −7.20736947999, −6.64877437759, −6.04347889941, −5.02803454913, −4.50862835068, −3.27477937083, 0, 0,
3.27477937083, 4.50862835068, 5.02803454913, 6.04347889941, 6.64877437759, 7.20736947999, 8.00098977891, 8.43899013961, 9.26423882619, 10.1489334004, 10.3253836772, 11.1205416798, 11.1708289929, 11.9135244006, 12.6214261523, 12.8115084431, 13.2192960322, 14.6517399504, 15.3214242661, 15.4626793990, 16.3347997060, 16.5663453795, 16.9697718502, 16.9988232526, 17.8739763542
