| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·9-s − 2·11-s + 3·13-s + 2·14-s + 16-s − 17-s + 2·18-s − 19-s + 2·22-s − 23-s + 2·25-s − 3·26-s − 3·27-s − 2·28-s − 5·29-s − 32-s + 34-s − 2·36-s + 7·37-s + 38-s + 4·41-s + 4·43-s − 2·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.832·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.426·22-s − 0.208·23-s + 2/5·25-s − 0.588·26-s − 0.577·27-s − 0.377·28-s − 0.928·29-s − 0.176·32-s + 0.171·34-s − 1/3·36-s + 1.15·37-s + 0.162·38-s + 0.624·41-s + 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3999421124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3999421124\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 40 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 14 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + T + 106 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−19.4640449309, −19.0846875955, −18.4561845702, −18.0313909197, −17.4443792418, −16.7606321293, −16.3440307938, −15.8647707463, −15.1622975313, −14.7106927042, −13.7862451196, −13.2575053833, −12.6860247727, −11.9349397730, −11.0756885544, −10.8441433884, −9.89975287019, −9.30065640034, −8.63556199344, −7.92261532941, −7.10539844128, −6.16278755263, −5.56183441474, −3.96320675063, −2.67343284092,
2.67343284092, 3.96320675063, 5.56183441474, 6.16278755263, 7.10539844128, 7.92261532941, 8.63556199344, 9.30065640034, 9.89975287019, 10.8441433884, 11.0756885544, 11.9349397730, 12.6860247727, 13.2575053833, 13.7862451196, 14.7106927042, 15.1622975313, 15.8647707463, 16.3440307938, 16.7606321293, 17.4443792418, 18.0313909197, 18.4561845702, 19.0846875955, 19.4640449309
