| L(s) = 1 | + 2-s + 2·3-s − 2·4-s − 2·5-s + 2·6-s + 3·7-s − 3·8-s + 9-s − 2·10-s − 5·11-s − 4·12-s − 13-s + 3·14-s − 4·15-s + 16-s + 5·17-s + 18-s − 5·19-s + 4·20-s + 6·21-s − 5·22-s + 8·23-s − 6·24-s + 2·25-s − 26-s + 2·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s + 1.13·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.50·11-s − 1.15·12-s − 0.277·13-s + 0.801·14-s − 1.03·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 1.14·19-s + 0.894·20-s + 1.30·21-s − 1.06·22-s + 1.66·23-s − 1.22·24-s + 2/5·25-s − 0.196·26-s + 0.384·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063660121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063660121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( 1 + T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 82 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 76 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−18.1224509119, −17.3976048601, −16.7329325002, −16.3316222321, −15.2702219431, −15.0595296804, −14.6843679621, −14.3230557454, −13.6412449146, −13.2135279380, −12.6453379533, −12.3284462243, −11.2988804125, −10.8497712810, −10.1967402857, −9.17493149096, −8.79981058738, −8.26757667195, −7.66172969083, −7.24977025857, −5.51348325422, −5.16191040547, −4.31844577816, −3.60480652087, −2.60296158052,
2.60296158052, 3.60480652087, 4.31844577816, 5.16191040547, 5.51348325422, 7.24977025857, 7.66172969083, 8.26757667195, 8.79981058738, 9.17493149096, 10.1967402857, 10.8497712810, 11.2988804125, 12.3284462243, 12.6453379533, 13.2135279380, 13.6412449146, 14.3230557454, 14.6843679621, 15.0595296804, 15.2702219431, 16.3316222321, 16.7329325002, 17.3976048601, 18.1224509119
