| L(s) = 1 | + 2·2-s + 2·4-s + 9-s + 2·13-s − 4·16-s + 8·17-s + 2·18-s + 25-s + 4·26-s − 8·32-s + 16·34-s + 2·36-s − 10·49-s + 2·50-s + 4·52-s − 12·53-s − 8·64-s + 16·68-s + 81-s + 20·89-s − 20·98-s + 2·100-s − 24·101-s − 24·106-s + 2·117-s + 3·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1/3·9-s + 0.554·13-s − 16-s + 1.94·17-s + 0.471·18-s + 1/5·25-s + 0.784·26-s − 1.41·32-s + 2.74·34-s + 1/3·36-s − 1.42·49-s + 0.282·50-s + 0.554·52-s − 1.64·53-s − 64-s + 1.94·68-s + 1/9·81-s + 2.11·89-s − 2.02·98-s + 1/5·100-s − 2.38·101-s − 2.33·106-s + 0.184·117-s + 3/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.783443290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.783443290\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−10.32710425179278583173374343686, −9.734036651812159865319714010221, −9.366535174229099991664797153639, −8.642861642349751575219520788241, −8.001409407726751008464122826538, −7.57650066630926266627463378984, −6.84694261571805877320048112308, −6.25366338754900606427181946660, −5.86116800229550123369788422302, −5.11090651406464491100348828758, −4.78603201462487435078510282892, −3.84437076432684289142922869440, −3.44385363211105369776585575975, −2.73402926377029765457846956606, −1.48049360260394175639927341072,
1.48049360260394175639927341072, 2.73402926377029765457846956606, 3.44385363211105369776585575975, 3.84437076432684289142922869440, 4.78603201462487435078510282892, 5.11090651406464491100348828758, 5.86116800229550123369788422302, 6.25366338754900606427181946660, 6.84694261571805877320048112308, 7.57650066630926266627463378984, 8.001409407726751008464122826538, 8.642861642349751575219520788241, 9.366535174229099991664797153639, 9.734036651812159865319714010221, 10.32710425179278583173374343686
