| L(s) = 1 | − 2-s − 3·5-s − 7-s − 8-s − 9-s + 3·10-s − 3·11-s + 3·13-s + 14-s − 16-s − 3·17-s + 18-s + 3·19-s + 3·22-s − 8·23-s − 25-s − 3·26-s − 5·29-s + 6·32-s + 3·34-s + 3·35-s − 37-s − 3·38-s + 3·40-s + 3·41-s − 4·43-s + 3·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 0.377·7-s − 0.353·8-s − 1/3·9-s + 0.948·10-s − 0.904·11-s + 0.832·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.688·19-s + 0.639·22-s − 1.66·23-s − 1/5·25-s − 0.588·26-s − 0.928·29-s + 1.06·32-s + 0.514·34-s + 0.507·35-s − 0.164·37-s − 0.486·38-s + 0.474·40-s + 0.468·41-s − 0.609·43-s + 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( 1 + T \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - T + 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T - 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−17.2078023914, −16.5432071058, −15.9837705923, −15.7595949692, −15.4227989670, −14.9137949515, −13.9727240105, −13.6734557298, −13.1369629509, −12.3159437482, −12.0272474641, −11.2641001746, −11.1399564314, −10.3118343571, −9.63189491675, −9.22188610233, −8.31256720373, −8.20390220796, −7.54990710676, −6.78797942665, −6.01447086439, −5.30365632565, −4.12339465788, −3.66442958906, −2.46208609884, 0,
2.46208609884, 3.66442958906, 4.12339465788, 5.30365632565, 6.01447086439, 6.78797942665, 7.54990710676, 8.20390220796, 8.31256720373, 9.22188610233, 9.63189491675, 10.3118343571, 11.1399564314, 11.2641001746, 12.0272474641, 12.3159437482, 13.1369629509, 13.6734557298, 13.9727240105, 14.9137949515, 15.4227989670, 15.7595949692, 15.9837705923, 16.5432071058, 17.2078023914
