| L(s) = 1 | + 3-s − 2·5-s − 3·7-s + 11-s − 3·13-s − 2·15-s + 17-s − 3·19-s − 3·21-s − 7·23-s + 2·25-s + 2·27-s − 10·29-s + 2·31-s + 33-s + 6·35-s − 3·37-s − 3·39-s − 43-s − 47-s − 6·49-s + 51-s + 53-s − 2·55-s − 3·57-s − 7·59-s + 9·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.13·7-s + 0.301·11-s − 0.832·13-s − 0.516·15-s + 0.242·17-s − 0.688·19-s − 0.654·21-s − 1.45·23-s + 2/5·25-s + 0.384·27-s − 1.85·29-s + 0.359·31-s + 0.174·33-s + 1.01·35-s − 0.493·37-s − 0.480·39-s − 0.152·43-s − 0.145·47-s − 6/7·49-s + 0.140·51-s + 0.137·53-s − 0.269·55-s − 0.397·57-s − 0.911·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 21 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 69 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 151 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−14.9359836462, −14.5778017765, −14.4625606642, −13.6074423839, −13.3287172429, −12.7514776540, −12.2438722914, −12.1411033206, −11.3556813037, −11.0261553408, −10.2295817802, −9.89854147069, −9.42823795351, −8.93786656467, −8.30712939340, −7.86058483950, −7.43140884182, −6.68662441190, −6.37726484701, −5.55835509737, −4.83747319427, −3.94415203947, −3.67127609420, −2.87523728458, −1.99961428021, 0,
1.99961428021, 2.87523728458, 3.67127609420, 3.94415203947, 4.83747319427, 5.55835509737, 6.37726484701, 6.68662441190, 7.43140884182, 7.86058483950, 8.30712939340, 8.93786656467, 9.42823795351, 9.89854147069, 10.2295817802, 11.0261553408, 11.3556813037, 12.1411033206, 12.2438722914, 12.7514776540, 13.3287172429, 13.6074423839, 14.4625606642, 14.5778017765, 14.9359836462
