L(s) = 1 | − 2·4-s + 3·5-s − 2·9-s + 6·13-s + 4·16-s − 3·17-s − 6·20-s − 25-s + 2·29-s + 4·36-s − 14·37-s − 12·41-s − 6·45-s − 4·49-s − 12·52-s + 18·53-s + 7·61-s − 8·64-s + 18·65-s + 6·68-s − 14·73-s + 12·80-s − 5·81-s − 9·85-s + 3·89-s + 7·97-s + 2·100-s + ⋯ |
L(s) = 1 | − 4-s + 1.34·5-s − 2/3·9-s + 1.66·13-s + 16-s − 0.727·17-s − 1.34·20-s − 1/5·25-s + 0.371·29-s + 2/3·36-s − 2.30·37-s − 1.87·41-s − 0.894·45-s − 4/7·49-s − 1.66·52-s + 2.47·53-s + 0.896·61-s − 64-s + 2.23·65-s + 0.727·68-s − 1.63·73-s + 1.34·80-s − 5/9·81-s − 0.976·85-s + 0.317·89-s + 0.710·97-s + 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8638295230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8638295230\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 5 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−11.92452692408347322735269236987, −11.61273287990120787720776932942, −10.62421186740291358053509556515, −10.31641110926210258202069372428, −9.752412479191050908402096379946, −8.903582143199867768728542514666, −8.710666107935939056437956579395, −8.224123461231545256743607430396, −7.02160059752068185550241650652, −6.31781713162194054768167062204, −5.63567096495448927563133329445, −5.25059973700108178017427319326, −4.11356927924001896534175820796, −3.28232247297491903361532119637, −1.80243843586560891979518056842,
1.80243843586560891979518056842, 3.28232247297491903361532119637, 4.11356927924001896534175820796, 5.25059973700108178017427319326, 5.63567096495448927563133329445, 6.31781713162194054768167062204, 7.02160059752068185550241650652, 8.224123461231545256743607430396, 8.710666107935939056437956579395, 8.903582143199867768728542514666, 9.752412479191050908402096379946, 10.31641110926210258202069372428, 10.62421186740291358053509556515, 11.61273287990120787720776932942, 11.92452692408347322735269236987