L(s) = 1 | + (0.279 − 0.0538i)2-s + (−0.853 + 0.341i)4-s + (−0.459 + 0.295i)8-s + (0.995 + 0.0950i)9-s + (−0.204 + 1.06i)11-s + (0.552 − 0.526i)16-s + (0.283 − 0.0270i)18-s + 0.307i·22-s + (0.580 + 0.814i)23-s + (−0.327 + 0.945i)25-s + (0.186 + 1.29i)29-s + (0.442 − 0.621i)32-s + (−0.881 + 0.258i)36-s + (−0.0535 + 0.560i)37-s + (0.983 − 1.53i)43-s + (−0.188 − 0.975i)44-s + ⋯ |
L(s) = 1 | + (0.279 − 0.0538i)2-s + (−0.853 + 0.341i)4-s + (−0.459 + 0.295i)8-s + (0.995 + 0.0950i)9-s + (−0.204 + 1.06i)11-s + (0.552 − 0.526i)16-s + (0.283 − 0.0270i)18-s + 0.307i·22-s + (0.580 + 0.814i)23-s + (−0.327 + 0.945i)25-s + (0.186 + 1.29i)29-s + (0.442 − 0.621i)32-s + (−0.881 + 0.258i)36-s + (−0.0535 + 0.560i)37-s + (0.983 − 1.53i)43-s + (−0.188 − 0.975i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9985614684\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9985614684\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 + (-0.580 - 0.814i)T \) |
good | 2 | \( 1 + (-0.279 + 0.0538i)T + (0.928 - 0.371i)T^{2} \) |
| 3 | \( 1 + (-0.995 - 0.0950i)T^{2} \) |
| 5 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 11 | \( 1 + (0.204 - 1.06i)T + (-0.928 - 0.371i)T^{2} \) |
| 13 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 19 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 29 | \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 37 | \( 1 + (0.0535 - 0.560i)T + (-0.981 - 0.189i)T^{2} \) |
| 41 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (-0.983 + 1.53i)T + (-0.415 - 0.909i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.46 + 0.356i)T + (0.888 - 0.458i)T^{2} \) |
| 59 | \( 1 + (0.0475 + 0.998i)T^{2} \) |
| 61 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 67 | \( 1 + (1.87 + 0.647i)T + (0.786 + 0.618i)T^{2} \) |
| 71 | \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.723 - 0.690i)T^{2} \) |
| 79 | \( 1 + (1.46 + 0.356i)T + (0.888 + 0.458i)T^{2} \) |
| 83 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 97 | \( 1 + (0.654 + 0.755i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02454972285752262824639652134, −9.341580814575998318833842887738, −8.639309586678537403087678005485, −7.43628663824263022403318158066, −7.14533113820612047014296404870, −5.64842310596065081938313210330, −4.86401639252533340314743311098, −4.12508945670596032944230139195, −3.14627823071091812250345643304, −1.63682824379856793164577380908,
0.953890809158487010274636955666, 2.71964960913044116100332124460, 4.01832140349784693729351359388, 4.54701221987763290917456556113, 5.70877189047463713907023057369, 6.31794362064405706683691431315, 7.46280391125527060376057057605, 8.400152351309864980838851895926, 9.058876133899583018256384500085, 9.985329616609483101931970837299