L(s) = 1 | + (−0.0980 − 0.995i)2-s + (−0.980 + 0.195i)4-s + (0.195 − 0.980i)5-s + (0.222 − 0.536i)7-s + (0.290 + 0.956i)8-s + (−0.382 − 0.923i)9-s + (−0.995 − 0.0980i)10-s + (−0.831 + 0.555i)11-s + (−0.247 − 1.24i)13-s + (−0.555 − 0.168i)14-s + (0.923 − 0.382i)16-s + (0.666 − 0.666i)17-s + (−0.881 + 0.471i)18-s + i·20-s + (0.634 + 0.773i)22-s + ⋯ |
L(s) = 1 | + (−0.0980 − 0.995i)2-s + (−0.980 + 0.195i)4-s + (0.195 − 0.980i)5-s + (0.222 − 0.536i)7-s + (0.290 + 0.956i)8-s + (−0.382 − 0.923i)9-s + (−0.995 − 0.0980i)10-s + (−0.831 + 0.555i)11-s + (−0.247 − 1.24i)13-s + (−0.555 − 0.168i)14-s + (0.923 − 0.382i)16-s + (0.666 − 0.666i)17-s + (−0.881 + 0.471i)18-s + i·20-s + (0.634 + 0.773i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7936462210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7936462210\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0980 + 0.995i)T \) |
| 5 | \( 1 + (-0.195 + 0.980i)T \) |
| 11 | \( 1 + (0.831 - 0.555i)T \) |
good | 3 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 7 | \( 1 + (-0.222 + 0.536i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (0.247 + 1.24i)T + (-0.923 + 0.382i)T^{2} \) |
| 17 | \( 1 + (-0.666 + 0.666i)T - iT^{2} \) |
| 19 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 23 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 - 1.66iT - T^{2} \) |
| 37 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 + (1.10 + 1.65i)T + (-0.382 + 0.923i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 59 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 71 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 73 | \( 1 + (0.0750 + 0.181i)T + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + (-1.24 + 0.247i)T + (0.923 - 0.382i)T^{2} \) |
| 89 | \( 1 + (1.81 + 0.750i)T + (0.707 + 0.707i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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
−8.496556504804114991922733679906, −7.83756762700215485150573179199, −7.04997151897242377498106702165, −5.59738077715160208998442706393, −5.24309767366861554230929230574, −4.43668671356172412120918577377, −3.45992847312238647246006149915, −2.73647749566261599740790498147, −1.47906965101684035875052213358, −0.48838983368566607671740854660,
1.88235674998933096405138719930, 2.82251884173274438413684058009, 3.89624780575128155436628458590, 4.87542602150186358631616595385, 5.62302361226425848207664421971, 6.17185580001757915035047962576, 6.93293166825334111915543412237, 7.939815484360417574551455163121, 8.012012308150801432759174779076, 9.059053309845063594548141014115