| L(s) = 1 | + 2-s − 4·7-s − 8-s + 3·11-s − 4·13-s − 4·14-s − 16-s + 6·17-s + 10·19-s + 3·22-s − 6·23-s − 4·26-s + 6·29-s − 2·31-s + 6·34-s + 8·37-s + 10·38-s − 3·41-s + 11·43-s − 6·46-s + 7·49-s + 12·53-s + 4·56-s + 6·58-s − 3·59-s + 10·61-s − 2·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·7-s − 0.353·8-s + 0.904·11-s − 1.10·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s + 2.29·19-s + 0.639·22-s − 1.25·23-s − 0.784·26-s + 1.11·29-s − 0.359·31-s + 1.02·34-s + 1.31·37-s + 1.62·38-s − 0.468·41-s + 1.67·43-s − 0.884·46-s + 49-s + 1.64·53-s + 0.534·56-s + 0.787·58-s − 0.390·59-s + 1.28·61-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.566821175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566821175\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.699351818900393595822257050111, −9.605848215609773402939544957879, −9.215091751176277948806727407592, −8.691404646559196459173988024168, −7.967605571141222859528505860505, −7.84144896634660928446929687852, −7.24627048369732391999283594904, −6.76619595709434143554395487968, −6.71248658222267425331884506450, −5.74680716443291661824498655398, −5.69838773170041315387570530160, −5.45702526952035240486692853150, −4.66996691897144852334544346830, −4.01939987469260157825986653599, −3.95731197952804616037349888435, −3.05693675681021222101574531057, −3.04061324049766734339289166861, −2.40078938437342417576423649212, −1.30698843814315406268481776280, −0.64604342364778283633835676747,
0.64604342364778283633835676747, 1.30698843814315406268481776280, 2.40078938437342417576423649212, 3.04061324049766734339289166861, 3.05693675681021222101574531057, 3.95731197952804616037349888435, 4.01939987469260157825986653599, 4.66996691897144852334544346830, 5.45702526952035240486692853150, 5.69838773170041315387570530160, 5.74680716443291661824498655398, 6.71248658222267425331884506450, 6.76619595709434143554395487968, 7.24627048369732391999283594904, 7.84144896634660928446929687852, 7.967605571141222859528505860505, 8.691404646559196459173988024168, 9.215091751176277948806727407592, 9.605848215609773402939544957879, 9.699351818900393595822257050111
