| L(s) = 1 | − 3-s − 2·5-s + 7-s + 9-s − 5·11-s + 3·13-s + 2·15-s + 3·17-s − 2·19-s − 21-s − 4·23-s + 2·25-s − 27-s − 5·29-s + 8·31-s + 5·33-s − 2·35-s − 6·37-s − 3·39-s + 2·41-s − 8·43-s − 2·45-s + 3·47-s + 7·49-s − 3·51-s + 2·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.832·13-s + 0.516·15-s + 0.727·17-s − 0.458·19-s − 0.218·21-s − 0.834·23-s + 2/5·25-s − 0.192·27-s − 0.928·29-s + 1.43·31-s + 0.870·33-s − 0.338·35-s − 0.986·37-s − 0.480·39-s + 0.312·41-s − 1.21·43-s − 0.298·45-s + 0.437·47-s + 49-s − 0.420·51-s + 0.274·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T - 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 102 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 150 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−14.1632762413, −13.8005623774, −13.3923424615, −12.8825860251, −12.4798049916, −11.9941652246, −11.6977884152, −11.1983922046, −10.6605575869, −10.4820449439, −9.96498569332, −9.39008666706, −8.58857617033, −8.21271931926, −7.98931589731, −7.31324088059, −6.95683456774, −6.12789033799, −5.68404511105, −5.21199392318, −4.51471721593, −4.02912828447, −3.31386240649, −2.53648421512, −1.43024670052, 0,
1.43024670052, 2.53648421512, 3.31386240649, 4.02912828447, 4.51471721593, 5.21199392318, 5.68404511105, 6.12789033799, 6.95683456774, 7.31324088059, 7.98931589731, 8.21271931926, 8.58857617033, 9.39008666706, 9.96498569332, 10.4820449439, 10.6605575869, 11.1983922046, 11.6977884152, 11.9941652246, 12.4798049916, 12.8825860251, 13.3923424615, 13.8005623774, 14.1632762413
