| L(s) = 1 | − 2-s − 7-s + 8-s + 4·13-s + 14-s − 16-s − 12·17-s + 4·19-s + 5·25-s − 4·26-s − 6·29-s + 4·31-s + 12·34-s + 4·37-s − 4·38-s + 6·41-s − 8·43-s − 12·47-s − 5·50-s − 12·53-s − 56-s + 6·58-s − 6·59-s − 8·61-s − 4·62-s + 64-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.377·7-s + 0.353·8-s + 1.10·13-s + 0.267·14-s − 1/4·16-s − 2.91·17-s + 0.917·19-s + 25-s − 0.784·26-s − 1.11·29-s + 0.718·31-s + 2.05·34-s + 0.657·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s − 1.75·47-s − 0.707·50-s − 1.64·53-s − 0.133·56-s + 0.787·58-s − 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7808564562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7808564562\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.891313454223295772593032602760, −9.535472416805858380599696163061, −9.144112326093391275416968878880, −8.808177010598245969218362258317, −8.608130871073625653206077853633, −7.981257120415105270255681726485, −7.69057115866209479965529724330, −7.15019160203012861630806528079, −6.58825241191196751708728243688, −6.31892139665696847393349166371, −6.16434002775949701438081814186, −5.08966268764508194729879453026, −5.01278522753931608490796923940, −4.26342269721344477796631016364, −4.01208733707856319603326521253, −3.13956001154375328248567534051, −2.87292493686573118849481064939, −1.94393534337681166868019465276, −1.48752664366294357642501161569, −0.44535638382632330893378840750,
0.44535638382632330893378840750, 1.48752664366294357642501161569, 1.94393534337681166868019465276, 2.87292493686573118849481064939, 3.13956001154375328248567534051, 4.01208733707856319603326521253, 4.26342269721344477796631016364, 5.01278522753931608490796923940, 5.08966268764508194729879453026, 6.16434002775949701438081814186, 6.31892139665696847393349166371, 6.58825241191196751708728243688, 7.15019160203012861630806528079, 7.69057115866209479965529724330, 7.981257120415105270255681726485, 8.608130871073625653206077853633, 8.808177010598245969218362258317, 9.144112326093391275416968878880, 9.535472416805858380599696163061, 9.891313454223295772593032602760
