| L(s) = 1 | + 4-s − 3·9-s + 4·13-s − 3·16-s − 5·19-s + 25-s − 3·36-s − 12·37-s + 8·43-s + 2·49-s + 4·52-s − 4·61-s − 7·64-s + 16·67-s + 4·73-s − 5·76-s − 16·79-s + 9·81-s − 4·97-s + 100-s + 24·103-s − 4·109-s − 12·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 9-s + 1.10·13-s − 3/4·16-s − 1.14·19-s + 1/5·25-s − 1/2·36-s − 1.97·37-s + 1.21·43-s + 2/7·49-s + 0.554·52-s − 0.512·61-s − 7/8·64-s + 1.95·67-s + 0.468·73-s − 0.573·76-s − 1.80·79-s + 81-s − 0.406·97-s + 1/10·100-s + 2.36·103-s − 0.383·109-s − 1.10·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8371177940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8371177940\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−12.38806137402189257544149202384, −11.75720866166289140797178003719, −11.12696178775400564714991072343, −10.87342505520780935626431209981, −10.23864961832214274208340490941, −9.276066313119881244608326892416, −8.659343963773707606473940933324, −8.387014710152601145504108157423, −7.37705423618747521171332225816, −6.65597620518539437284603188274, −6.09935801467255744090931321094, −5.35339473826414865763119861489, −4.28871303643889741765932991178, −3.30242673427684843025859892223, −2.16256815209145760969795398783,
2.16256815209145760969795398783, 3.30242673427684843025859892223, 4.28871303643889741765932991178, 5.35339473826414865763119861489, 6.09935801467255744090931321094, 6.65597620518539437284603188274, 7.37705423618747521171332225816, 8.387014710152601145504108157423, 8.659343963773707606473940933324, 9.276066313119881244608326892416, 10.23864961832214274208340490941, 10.87342505520780935626431209981, 11.12696178775400564714991072343, 11.75720866166289140797178003719, 12.38806137402189257544149202384
