| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s + 8-s − 2·10-s − 2·11-s − 2·12-s + 4·15-s + 16-s + 2·17-s + 2·19-s − 2·20-s − 2·22-s + 4·23-s − 2·24-s − 2·25-s + 2·27-s − 2·29-s + 4·30-s + 4·31-s + 32-s + 4·33-s + 2·34-s + 2·37-s + 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s + 0.353·8-s − 0.632·10-s − 0.603·11-s − 0.577·12-s + 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s − 0.408·24-s − 2/5·25-s + 0.384·27-s − 0.371·29-s + 0.730·30-s + 0.718·31-s + 0.176·32-s + 0.696·33-s + 0.342·34-s + 0.328·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5389915528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5389915528\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 18 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−19.6599673796, −18.8257037435, −18.6538819312, −17.7314598952, −17.1368418682, −16.8899209644, −15.9977541843, −15.7799166504, −15.1674772702, −14.4724742836, −13.8501037621, −13.0775734455, −12.5401203308, −11.8438109868, −11.4913322601, −10.9960866060, −10.2918633816, −9.41372745718, −8.26116678365, −7.64511681208, −6.82215132425, −5.85901766872, −5.29163110264, −4.35018172500, −3.12239061990,
3.12239061990, 4.35018172500, 5.29163110264, 5.85901766872, 6.82215132425, 7.64511681208, 8.26116678365, 9.41372745718, 10.2918633816, 10.9960866060, 11.4913322601, 11.8438109868, 12.5401203308, 13.0775734455, 13.8501037621, 14.4724742836, 15.1674772702, 15.7799166504, 15.9977541843, 16.8899209644, 17.1368418682, 17.7314598952, 18.6538819312, 18.8257037435, 19.6599673796
