| L(s) = 1 | + 4-s + 5-s − 2·9-s + 4·11-s − 3·16-s + 20-s + 8·23-s − 2·25-s + 16·31-s − 2·36-s − 12·37-s + 4·44-s − 2·45-s − 8·47-s + 49-s + 4·53-s + 4·55-s − 8·59-s − 7·64-s + 16·67-s − 3·80-s − 5·81-s − 12·89-s + 8·92-s + 4·97-s − 8·99-s − 2·100-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.447·5-s − 2/3·9-s + 1.20·11-s − 3/4·16-s + 0.223·20-s + 1.66·23-s − 2/5·25-s + 2.87·31-s − 1/3·36-s − 1.97·37-s + 0.603·44-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.549·53-s + 0.539·55-s − 1.04·59-s − 7/8·64-s + 1.95·67-s − 0.335·80-s − 5/9·81-s − 1.27·89-s + 0.834·92-s + 0.406·97-s − 0.804·99-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.510756486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.510756486\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.51377562579191192419378498884, −10.03307073252113038758637688342, −9.419600909250353776412496770470, −8.969656321983476769069515413708, −8.496760051636488016145458898546, −7.949065157743774593963273798095, −7.00925633495393877272481390137, −6.64548998507080451471956617386, −6.34800515038414072042156683594, −5.45420315327685117150039189890, −4.89183682654501834323686964315, −4.11710921769059857870196378484, −3.17645149887120785882158528523, −2.52320608208698457831371682398, −1.39200507089519553660797674573,
1.39200507089519553660797674573, 2.52320608208698457831371682398, 3.17645149887120785882158528523, 4.11710921769059857870196378484, 4.89183682654501834323686964315, 5.45420315327685117150039189890, 6.34800515038414072042156683594, 6.64548998507080451471956617386, 7.00925633495393877272481390137, 7.949065157743774593963273798095, 8.496760051636488016145458898546, 8.969656321983476769069515413708, 9.419600909250353776412496770470, 10.03307073252113038758637688342, 10.51377562579191192419378498884
