| L(s) = 1 | − 2-s − 2·3-s − 3·5-s + 2·6-s − 4·7-s + 8-s − 3·9-s + 3·10-s − 3·11-s + 4·14-s + 6·15-s − 16-s − 6·17-s + 3·18-s − 5·19-s + 8·21-s + 3·22-s − 2·24-s + 4·25-s + 14·27-s + 3·29-s − 6·30-s − 9·31-s + 6·33-s + 6·34-s + 12·35-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1.34·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s − 9-s + 0.948·10-s − 0.904·11-s + 1.06·14-s + 1.54·15-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 1.14·19-s + 1.74·21-s + 0.639·22-s − 0.408·24-s + 4/5·25-s + 2.69·27-s + 0.557·29-s − 1.09·30-s − 1.61·31-s + 1.04·33-s + 1.02·34-s + 2.02·35-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 33 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 67 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 121 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 60 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T - 105 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−16.0221711185, −15.5342970694, −14.9897584332, −14.6635859038, −13.8099113219, −13.2907075077, −12.8523853416, −12.4418385025, −11.9521338934, −11.3387256738, −11.0645280668, −10.6807001131, −10.1971453750, −9.45577863031, −8.77312346071, −8.61883253709, −7.95635897687, −7.25503634354, −6.73099839433, −6.12036715031, −5.72019169432, −4.78117523089, −4.24511295137, −3.26424045735, −2.58974577135, 0, 0,
2.58974577135, 3.26424045735, 4.24511295137, 4.78117523089, 5.72019169432, 6.12036715031, 6.73099839433, 7.25503634354, 7.95635897687, 8.61883253709, 8.77312346071, 9.45577863031, 10.1971453750, 10.6807001131, 11.0645280668, 11.3387256738, 11.9521338934, 12.4418385025, 12.8523853416, 13.2907075077, 13.8099113219, 14.6635859038, 14.9897584332, 15.5342970694, 16.0221711185
