L(s) = 1 | + (−0.955 + 0.294i)7-s + (1.81 + 0.414i)13-s + (−0.510 + 0.294i)19-s + (0.955 + 0.294i)25-s + (0.975 + 0.563i)31-s + (−0.123 − 0.0841i)37-s + (0.914 + 1.14i)43-s + (0.826 − 0.563i)49-s + (−0.488 + 0.716i)61-s + (0.988 − 1.71i)67-s + (−0.587 + 1.90i)73-s + (−0.826 − 1.43i)79-s + (−1.85 + 0.139i)91-s − 1.94i·97-s + (−0.202 + 1.34i)103-s + ⋯ |
L(s) = 1 | + (−0.955 + 0.294i)7-s + (1.81 + 0.414i)13-s + (−0.510 + 0.294i)19-s + (0.955 + 0.294i)25-s + (0.975 + 0.563i)31-s + (−0.123 − 0.0841i)37-s + (0.914 + 1.14i)43-s + (0.826 − 0.563i)49-s + (−0.488 + 0.716i)61-s + (0.988 − 1.71i)67-s + (−0.587 + 1.90i)73-s + (−0.826 − 1.43i)79-s + (−1.85 + 0.139i)91-s − 1.94i·97-s + (−0.202 + 1.34i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.089597769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.089597769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.955 - 0.294i)T \) |
good | 5 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 11 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 13 | \( 1 + (-1.81 - 0.414i)T + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 19 | \( 1 + (0.510 - 0.294i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.988 + 0.149i)T^{2} \) |
| 29 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.975 - 0.563i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.123 + 0.0841i)T + (0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (-0.914 - 1.14i)T + (-0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 53 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 59 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 61 | \( 1 + (0.488 - 0.716i)T + (-0.365 - 0.930i)T^{2} \) |
| 67 | \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.587 - 1.90i)T + (-0.826 - 0.563i)T^{2} \) |
| 79 | \( 1 + (0.826 + 1.43i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 97 | \( 1 + 1.94iT - T^{2} \) |
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
−9.423282713563092939196015140071, −8.792034041245095274606327704949, −8.170237068506724698088807464966, −6.97575889865620406736845733898, −6.33176769038514206868756224919, −5.73965726963788637150148330585, −4.50096995834584605858117878017, −3.60825410266652176689266513474, −2.76935023174560352958547014707, −1.32402336706378373783659359082,
0.993839710027635202613149549972, 2.59858640958885467647128549044, 3.55000175807754458354544858199, 4.28733769170551877340445281252, 5.53472141665947407518960640543, 6.32200678918440947954269025908, 6.85293260577674644039595012225, 7.957970859915843833448638079092, 8.692275670586855025054433759086, 9.330750922646498099133373597261