L(s) = 1 | + (−0.109 − 1.40i)2-s + (−1.97 + 0.307i)4-s + (3.44 − 1.99i)5-s + (0.212 + 2.63i)7-s + (0.649 + 2.75i)8-s + (−3.18 − 4.64i)10-s + (0.936 − 1.62i)11-s + 1.05·13-s + (3.69 − 0.587i)14-s + (3.81 − 1.21i)16-s + (2.30 + 1.33i)17-s + (0.628 − 0.362i)19-s + (−6.20 + 4.99i)20-s + (−2.38 − 1.14i)22-s + (3.00 + 5.20i)23-s + ⋯ |
L(s) = 1 | + (−0.0771 − 0.997i)2-s + (−0.988 + 0.153i)4-s + (1.54 − 0.890i)5-s + (0.0804 + 0.996i)7-s + (0.229 + 0.973i)8-s + (−1.00 − 1.46i)10-s + (0.282 − 0.489i)11-s + 0.293·13-s + (0.987 − 0.157i)14-s + (0.952 − 0.304i)16-s + (0.558 + 0.322i)17-s + (0.144 − 0.0831i)19-s + (−1.38 + 1.11i)20-s + (−0.509 − 0.243i)22-s + (0.626 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39930 - 1.17769i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39930 - 1.17769i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.109 + 1.40i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.212 - 2.63i)T \) |
good | 5 | \( 1 + (-3.44 + 1.99i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.936 + 1.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + (-2.30 - 1.33i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.628 + 0.362i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.00 - 5.20i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.09iT - 29T^{2} \) |
| 31 | \( 1 + (3.48 + 2.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.48 - 7.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.19iT - 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 + (-2.03 - 3.52i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (11.2 + 6.48i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.56 + 7.90i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.21 + 9.02i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.00 + 2.88i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (7.37 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.92 - 2.84i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 + (-8.80 + 5.08i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.03T + 97T^{2} \) |
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
−9.847284483443191575795897085955, −9.522914134974167513505592801228, −8.766131791291302063406696672665, −8.055209745197265625416813671820, −6.19783716696381966844724678119, −5.54551849499490831987266376529, −4.81919576604562825746482461168, −3.34336085223376999006493928892, −2.16609323747609188204707127962, −1.25059542712530604939566680645,
1.37992524110029741813017851871, 3.04801596246433218108472441591, 4.36376057564028438810096396762, 5.43329501654238150454954495160, 6.27608000313181184963282256029, 6.99264690578156754579772318917, 7.60151112887899518407674124468, 8.988416928596944514911310867990, 9.564367956732565910599989569140, 10.44936515609579028263465133998