L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s + 1.99·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.999 − 1.73i)12-s − 2·13-s + (0.500 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)17-s + (−0.499 + 0.866i)18-s + (−4 − 6.92i)19-s + (4 − 3.46i)21-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s + 0.816·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.288 − 0.499i)12-s − 0.554·13-s + (0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)17-s + (−0.117 + 0.204i)18-s + (−0.917 − 1.58i)19-s + (0.872 − 0.755i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.5 + 0.866i)T \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−10.70138519783764565186964169154, −10.36719368858153754106295936698, −9.447538433637964827813887110156, −8.626837508856333860725879211942, −7.19549461785947210748558553879, −6.12833913364576606119931117135, −4.71340149688995589082631396094, −3.97606473678978987500593465775, −2.52147167459023676507551059403, 0,
1.95644007893676770151609610298, 3.89127764587166431167854657345, 5.71416989088891006820420106006, 6.09227595154290483609847831744, 7.24638800259019867737969740239, 7.79471734969401737805172119337, 9.239073642448480033914322638940, 9.816356773338674004119608832666, 11.09137872051302255188412038105