| L(s) = 1 | − 3-s − 4-s + 2·5-s − 7-s + 12-s − 13-s − 2·15-s − 3·16-s − 7·17-s − 2·20-s + 21-s + 7·23-s + 2·25-s + 4·27-s + 28-s − 4·29-s + 31-s − 2·35-s − 12·37-s + 39-s + 8·41-s + 4·43-s + 10·47-s + 3·48-s + 5·49-s + 7·51-s + 52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.288·12-s − 0.277·13-s − 0.516·15-s − 3/4·16-s − 1.69·17-s − 0.447·20-s + 0.218·21-s + 1.45·23-s + 2/5·25-s + 0.769·27-s + 0.188·28-s − 0.742·29-s + 0.179·31-s − 0.338·35-s − 1.97·37-s + 0.160·39-s + 1.24·41-s + 0.609·43-s + 1.45·47-s + 0.433·48-s + 5/7·49-s + 0.980·51-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5166333764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5166333764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 154 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 130 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.0543889807, −18.4847400127, −17.8521407421, −17.5258273518, −17.1812758898, −16.5180934753, −15.8294171586, −15.3989395807, −14.6394085294, −13.9298090701, −13.4999518743, −12.9937828154, −12.4492157850, −11.6272411997, −10.8721090522, −10.5535484208, −9.60097056720, −8.97236501167, −8.71919614220, −7.18220895634, −6.77459577894, −5.82810065846, −5.08532681663, −4.20103980167, −2.51532317468,
2.51532317468, 4.20103980167, 5.08532681663, 5.82810065846, 6.77459577894, 7.18220895634, 8.71919614220, 8.97236501167, 9.60097056720, 10.5535484208, 10.8721090522, 11.6272411997, 12.4492157850, 12.9937828154, 13.4999518743, 13.9298090701, 14.6394085294, 15.3989395807, 15.8294171586, 16.5180934753, 17.1812758898, 17.5258273518, 17.8521407421, 18.4847400127, 19.0543889807
