| L(s) = 1 | − 2-s − 3·5-s − 4·7-s + 8-s + 3·10-s − 8·13-s + 4·14-s − 16-s − 6·17-s + 4·19-s − 6·23-s + 5·25-s + 8·26-s − 6·29-s − 8·31-s + 6·34-s + 12·35-s − 8·37-s − 4·38-s − 3·40-s − 12·41-s + 16·43-s + 6·46-s − 6·47-s + 9·49-s − 5·50-s − 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.34·5-s − 1.51·7-s + 0.353·8-s + 0.948·10-s − 2.21·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 25-s + 1.56·26-s − 1.11·29-s − 1.43·31-s + 1.02·34-s + 2.02·35-s − 1.31·37-s − 0.648·38-s − 0.474·40-s − 1.87·41-s + 2.43·43-s + 0.884·46-s − 0.875·47-s + 9/7·49-s − 0.707·50-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.08074060311665111223154391802, −10.71548012351716141722099535006, −9.856014431045838114836370144792, −9.817038568727010083127578875664, −9.463959626638406117033354024069, −8.870544233028312757080424898694, −8.358372067608522144412587162785, −7.896234967844326251086013308468, −7.19012930553513925486677764746, −7.11774526589412972998114004468, −6.76116813932201965995115383970, −5.77496015210104947812086959524, −5.27232366580793983725540357814, −4.60757573440944094998790977367, −3.93197969139871681081184968254, −3.55703412685945190466085042814, −2.72940543881168905302787287823, −1.99864911447537630592747978632, 0, 0,
1.99864911447537630592747978632, 2.72940543881168905302787287823, 3.55703412685945190466085042814, 3.93197969139871681081184968254, 4.60757573440944094998790977367, 5.27232366580793983725540357814, 5.77496015210104947812086959524, 6.76116813932201965995115383970, 7.11774526589412972998114004468, 7.19012930553513925486677764746, 7.896234967844326251086013308468, 8.358372067608522144412587162785, 8.870544233028312757080424898694, 9.463959626638406117033354024069, 9.817038568727010083127578875664, 9.856014431045838114836370144792, 10.71548012351716141722099535006, 11.08074060311665111223154391802
