L(s) = 1 | + 4-s − 2·11-s − 3·16-s − 3·19-s − 5·25-s + 12·29-s + 3·31-s − 9·41-s − 2·44-s − 5·49-s + 4·59-s − 21·61-s − 7·64-s + 22·71-s − 3·76-s + 12·79-s − 9·81-s + 8·89-s − 5·100-s + 4·101-s + 21·109-s + 12·116-s − 3·121-s + 3·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.603·11-s − 3/4·16-s − 0.688·19-s − 25-s + 2.22·29-s + 0.538·31-s − 1.40·41-s − 0.301·44-s − 5/7·49-s + 0.520·59-s − 2.68·61-s − 7/8·64-s + 2.61·71-s − 0.344·76-s + 1.35·79-s − 81-s + 0.847·89-s − 1/2·100-s + 0.398·101-s + 2.01·109-s + 1.11·116-s − 0.272·121-s + 0.269·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8550314191\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8550314191\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_2$ | \( 1 + p T^{2} \) |
| 179 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
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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
−12.18918167804562075041683901287, −11.91174558478368732921328236765, −11.15784032260619640773533614357, −10.64410761568902312948617043340, −10.09824820839617711680801197799, −9.486285470594352926661943054562, −8.598744632548457290278597687345, −8.161520469114403666133450142719, −7.42552510871024539328538624858, −6.55242182696352198698522228540, −6.24250290881522639965665078254, −5.08092392375524683764994471481, −4.44707173903413455078484909830, −3.19927072892018814112145833709, −2.15903857527108589111778351413,
2.15903857527108589111778351413, 3.19927072892018814112145833709, 4.44707173903413455078484909830, 5.08092392375524683764994471481, 6.24250290881522639965665078254, 6.55242182696352198698522228540, 7.42552510871024539328538624858, 8.161520469114403666133450142719, 8.598744632548457290278597687345, 9.486285470594352926661943054562, 10.09824820839617711680801197799, 10.64410761568902312948617043340, 11.15784032260619640773533614357, 11.91174558478368732921328236765, 12.18918167804562075041683901287