| L(s) = 1 | + 2-s + 3·3-s − 4-s − 4·5-s + 3·6-s + 5·7-s − 8-s + 3·9-s − 4·10-s − 3·11-s − 3·12-s + 2·13-s + 5·14-s − 12·15-s − 16-s + 2·17-s + 3·18-s − 2·19-s + 4·20-s + 15·21-s − 3·22-s + 2·23-s − 3·24-s + 2·25-s + 2·26-s − 5·28-s − 12·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.78·5-s + 1.22·6-s + 1.88·7-s − 0.353·8-s + 9-s − 1.26·10-s − 0.904·11-s − 0.866·12-s + 0.554·13-s + 1.33·14-s − 3.09·15-s − 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.894·20-s + 3.27·21-s − 0.639·22-s + 0.417·23-s − 0.612·24-s + 2/5·25-s + 0.392·26-s − 0.944·28-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10693 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10693 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722438891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722438891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−16.1892288014, −15.7159151158, −15.1422419847, −15.0693535709, −14.5224447253, −13.9795262067, −13.8073233945, −13.2904115004, −12.6191494050, −11.9362311444, −11.6668009653, −10.8145231408, −10.7173409989, −9.56332674948, −8.69568671187, −8.63066378577, −7.92228282646, −7.81910395524, −7.25087408024, −5.85138298935, −4.74199315541, −4.68908025029, −3.72938004472, −3.29762459285, −2.07228699821,
2.07228699821, 3.29762459285, 3.72938004472, 4.68908025029, 4.74199315541, 5.85138298935, 7.25087408024, 7.81910395524, 7.92228282646, 8.63066378577, 8.69568671187, 9.56332674948, 10.7173409989, 10.8145231408, 11.6668009653, 11.9362311444, 12.6191494050, 13.2904115004, 13.8073233945, 13.9795262067, 14.5224447253, 15.0693535709, 15.1422419847, 15.7159151158, 16.1892288014
