| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 3·8-s − 2·9-s + 10-s + 4·11-s + 12-s − 3·13-s − 14-s − 15-s + 16-s − 5·17-s + 2·18-s + 5·19-s − 20-s + 21-s − 4·22-s − 3·23-s − 3·24-s + 25-s + 3·26-s − 2·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.625·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s − 0.384·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5289763116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5289763116\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 101 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 10 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 5 T - 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 198 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−19.0214118579, −18.0306070202, −17.6428411512, −17.5004010751, −16.6357569253, −16.0681717028, −15.5748428210, −14.8887790826, −14.5543092177, −13.9920125972, −13.3870195377, −12.3434995109, −11.8247165348, −11.5663976229, −10.8505827130, −9.90786186516, −9.25489123825, −8.83955908714, −8.28713153605, −7.43323372881, −6.76917756928, −5.93457142870, −4.74853285170, −3.57009398076, −2.42030115863,
2.42030115863, 3.57009398076, 4.74853285170, 5.93457142870, 6.76917756928, 7.43323372881, 8.28713153605, 8.83955908714, 9.25489123825, 9.90786186516, 10.8505827130, 11.5663976229, 11.8247165348, 12.3434995109, 13.3870195377, 13.9920125972, 14.5543092177, 14.8887790826, 15.5748428210, 16.0681717028, 16.6357569253, 17.5004010751, 17.6428411512, 18.0306070202, 19.0214118579
