L(s) = 1 | + (0.733 − 0.680i)7-s + (−0.290 + 0.0663i)13-s + (1.17 − 0.680i)19-s + (−0.733 − 0.680i)25-s + (1.72 + 0.997i)31-s + (−0.123 − 1.64i)37-s + (−1.19 + 1.49i)43-s + (0.0747 − 0.997i)49-s + (0.865 − 0.0648i)61-s + (−0.365 + 0.632i)67-s + (0.766 − 0.825i)73-s + (−0.0747 − 0.129i)79-s + (−0.167 + 0.246i)91-s + 1.94i·97-s + (0.548 + 0.215i)103-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)7-s + (−0.290 + 0.0663i)13-s + (1.17 − 0.680i)19-s + (−0.733 − 0.680i)25-s + (1.72 + 0.997i)31-s + (−0.123 − 1.64i)37-s + (−1.19 + 1.49i)43-s + (0.0747 − 0.997i)49-s + (0.865 − 0.0648i)61-s + (−0.365 + 0.632i)67-s + (0.766 − 0.825i)73-s + (−0.0747 − 0.129i)79-s + (−0.167 + 0.246i)91-s + 1.94i·97-s + (0.548 + 0.215i)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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.234810305\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.234810305\) |
\(L(1)\) |
|
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 | \( 1 + (-0.733 + 0.680i)T \) |
good | 5 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (0.826 + 0.563i)T^{2} \) |
| 13 | \( 1 + (0.290 - 0.0663i)T + (0.900 - 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (-1.17 + 0.680i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-1.72 - 0.997i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.123 + 1.64i)T + (-0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (-0.988 - 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (-0.865 + 0.0648i)T + (0.988 - 0.149i)T^{2} \) |
| 67 | \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.766 + 0.825i)T + (-0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 97 | \( 1 - 1.94iT - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.529229448341835332260341091734, −8.531948986353964913766185922140, −7.83809334844241547926134500708, −7.14746585351734644816208800174, −6.31222941964213960508474724466, −5.18132579270635553157223875424, −4.58475324359323394112303627777, −3.56497684144339151846282817276, −2.43204871924408024025778435751, −1.11219254727807093144761130914,
1.46106891732136234811089879572, 2.57817256658125249591440327156, 3.63339628918369463797939553629, 4.77455991568665451288862846589, 5.43432832683261127388011543105, 6.25924633511470897511475498376, 7.30201878511433081899063394356, 8.063337092514712810207827344128, 8.613257707707270488623227041730, 9.701580440950616233246559500343