| L(s) = 1 | − 2-s − 4-s − 6·5-s + 3·8-s − 5·9-s + 6·10-s + 6·13-s − 16-s − 12·17-s + 5·18-s + 6·20-s + 17·25-s − 6·26-s − 12·29-s − 5·32-s + 12·34-s + 5·36-s − 4·37-s − 18·40-s − 20·41-s + 30·45-s − 13·49-s − 17·50-s − 6·52-s + 16·53-s + 12·58-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 2.68·5-s + 1.06·8-s − 5/3·9-s + 1.89·10-s + 1.66·13-s − 1/4·16-s − 2.91·17-s + 1.17·18-s + 1.34·20-s + 17/5·25-s − 1.17·26-s − 2.22·29-s − 0.883·32-s + 2.05·34-s + 5/6·36-s − 0.657·37-s − 2.84·40-s − 3.12·41-s + 4.47·45-s − 1.85·49-s − 2.40·50-s − 0.832·52-s + 2.19·53-s + 1.57·58-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p T^{2} )^{2} \) |
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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
−8.806695473952422658633000158062, −8.564424981103890785422457175495, −8.282431792555002607496667489786, −7.903372833693644985808766966158, −7.12147698344619796195085066396, −6.86805667484759342647357469603, −6.09922127840546731746771292105, −5.29917071437467895295333587611, −4.70019585546845455212568823793, −3.98638135363332626512942033326, −3.76592143771251579951082054976, −3.18285249010353951279255341401, −1.87107040460257588490750995471, 0, 0,
1.87107040460257588490750995471, 3.18285249010353951279255341401, 3.76592143771251579951082054976, 3.98638135363332626512942033326, 4.70019585546845455212568823793, 5.29917071437467895295333587611, 6.09922127840546731746771292105, 6.86805667484759342647357469603, 7.12147698344619796195085066396, 7.903372833693644985808766966158, 8.282431792555002607496667489786, 8.564424981103890785422457175495, 8.806695473952422658633000158062
