| L(s) = 1 | + (−0.826 + 0.563i)2-s + (0.733 − 0.680i)3-s + (0.365 − 0.930i)4-s + (2.66 + 0.820i)5-s + (−0.222 + 0.974i)6-s + (0.800 + 2.52i)7-s + (0.222 + 0.974i)8-s + (0.0747 − 0.997i)9-s + (−2.66 + 0.820i)10-s + (0.0778 + 1.03i)11-s + (−0.365 − 0.930i)12-s + (−4.86 − 2.34i)13-s + (−2.08 − 1.63i)14-s + (2.50 − 1.20i)15-s + (−0.733 − 0.680i)16-s + (7.47 + 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.584 + 0.398i)2-s + (0.423 − 0.392i)3-s + (0.182 − 0.465i)4-s + (1.19 + 0.367i)5-s + (−0.0908 + 0.398i)6-s + (0.302 + 0.953i)7-s + (0.0786 + 0.344i)8-s + (0.0249 − 0.332i)9-s + (−0.841 + 0.259i)10-s + (0.0234 + 0.313i)11-s + (−0.105 − 0.268i)12-s + (−1.35 − 0.650i)13-s + (−0.556 − 0.436i)14-s + (0.647 − 0.311i)15-s + (−0.183 − 0.170i)16-s + (1.81 + 0.273i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30638 + 0.354090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30638 + 0.354090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.826 - 0.563i)T \) |
| 3 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.800 - 2.52i)T \) |
| good | 5 | \( 1 + (-2.66 - 0.820i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (-0.0778 - 1.03i)T + (-10.8 + 1.63i)T^{2} \) |
| 13 | \( 1 + (4.86 + 2.34i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-7.47 - 1.12i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.0654 - 0.113i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.640 + 0.0965i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (1.18 - 1.48i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.56 - 2.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.97 + 10.1i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (0.342 + 1.50i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (1.37 - 6.02i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-7.34 + 5.00i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-2.71 + 6.92i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (10.5 - 3.26i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (1.68 + 4.30i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-1.24 - 2.16i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.16 + 2.71i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (12.4 + 8.49i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-2.81 + 4.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (12.4 - 6.00i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.0443 + 0.591i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
| 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
−12.06439065356991321522613008458, −10.52186989309359137124079733545, −9.821976778113570827624788636741, −9.091834647515896521453718556758, −7.969716733561036021147936562063, −7.15753921569942930311311116164, −5.89535781213999082306674998044, −5.25820607805463164229050145402, −2.87561554926051351818088253537, −1.80826070947044072299924284420,
1.46408955985280220534409442066, 2.91450238202253086754071579552, 4.41336886689631497249028518327, 5.57457070579149645206072732658, 7.09779167946912941968453874784, 7.977223547523987215042326224786, 9.162605982705316429743027733983, 9.913451518116247319836244363190, 10.30102945154487959522702321042, 11.59855003929108248365970803718
