| L(s) = 1 | − 3-s − 5·7-s + 9-s + 2·11-s + 13-s − 2·17-s − 19-s + 5·21-s − 10·23-s + 25-s − 27-s − 4·31-s − 2·33-s − 37-s − 39-s − 8·41-s − 4·43-s + 11·49-s + 2·51-s − 4·53-s + 57-s + 8·59-s + 5·61-s − 5·63-s + 5·67-s + 10·69-s − 22·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s − 0.485·17-s − 0.229·19-s + 1.09·21-s − 2.08·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s − 0.348·33-s − 0.164·37-s − 0.160·39-s − 1.24·41-s − 0.609·43-s + 11/7·49-s + 0.280·51-s − 0.549·53-s + 0.132·57-s + 1.04·59-s + 0.640·61-s − 0.629·63-s + 0.610·67-s + 1.20·69-s − 2.61·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 5 T + 106 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 24 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 70 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−15.6767586690, −15.0521486342, −14.4837948602, −14.0181089352, −13.4471958374, −12.9884387212, −12.7920519994, −12.1251669396, −11.7081834971, −11.3748859747, −10.4975591888, −10.0848725577, −9.92329543116, −9.14933255526, −8.79006016606, −8.09677153379, −7.30389943030, −6.75900134747, −6.32754210755, −5.96142298096, −5.23820561080, −4.23564211599, −3.76388131546, −3.03751652909, −1.85975146490, 0,
1.85975146490, 3.03751652909, 3.76388131546, 4.23564211599, 5.23820561080, 5.96142298096, 6.32754210755, 6.75900134747, 7.30389943030, 8.09677153379, 8.79006016606, 9.14933255526, 9.92329543116, 10.0848725577, 10.4975591888, 11.3748859747, 11.7081834971, 12.1251669396, 12.7920519994, 12.9884387212, 13.4471958374, 14.0181089352, 14.4837948602, 15.0521486342, 15.6767586690
