| L(s) = 1 | + (0.152 + 1.40i)2-s + (−0.453 − 0.891i)3-s + (−1.95 + 0.427i)4-s + (1.07 − 1.96i)5-s + (1.18 − 0.773i)6-s + (−0.868 − 0.868i)7-s + (−0.898 − 2.68i)8-s + (−0.587 + 0.809i)9-s + (2.92 + 1.21i)10-s + (−2.63 − 3.62i)11-s + (1.26 + 1.54i)12-s + (−0.530 − 3.34i)13-s + (1.08 − 1.35i)14-s + (−2.23 − 0.0672i)15-s + (3.63 − 1.67i)16-s + (5.52 + 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.107 + 0.994i)2-s + (−0.262 − 0.514i)3-s + (−0.976 + 0.213i)4-s + (0.480 − 0.876i)5-s + (0.483 − 0.315i)6-s + (−0.328 − 0.328i)7-s + (−0.317 − 0.948i)8-s + (−0.195 + 0.269i)9-s + (0.923 + 0.383i)10-s + (−0.794 − 1.09i)11-s + (0.366 + 0.446i)12-s + (−0.147 − 0.928i)13-s + (0.291 − 0.361i)14-s + (−0.577 − 0.0173i)15-s + (0.908 − 0.417i)16-s + (1.33 + 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.914131 - 0.396498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.914131 - 0.396498i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.152 - 1.40i)T \) |
| 3 | \( 1 + (0.453 + 0.891i)T \) |
| 5 | \( 1 + (-1.07 + 1.96i)T \) |
| good | 7 | \( 1 + (0.868 + 0.868i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.63 + 3.62i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.530 + 3.34i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.52 - 2.81i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.47 + 4.52i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.748 - 4.72i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-2.21 - 0.720i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-9.32 + 3.02i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (8.45 - 1.33i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.25 - 1.63i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.74 - 5.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (4.24 - 2.16i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.643 + 0.327i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.305 - 0.222i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.68 + 1.95i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-3.01 + 5.92i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-4.75 - 1.54i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-3.39 - 0.537i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.521 + 1.60i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-13.5 - 6.92i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (0.948 + 1.30i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.55 + 10.9i)T + (-57.0 + 78.4i)T^{2} \) |
| 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
−11.94354843474670160936581280292, −10.45538489558262134683983884290, −9.631700593935123502586815361487, −8.293512207577886834859303939724, −7.977675680842150748696830601510, −6.56203050445980145988132168147, −5.67390012456217470269354891913, −4.97488100043537960436429093843, −3.27581180371169914276699615565, −0.75499972484596620727975421056,
2.14826424024004644533552551317, 3.26665373317397633052480749819, 4.61344722217131461409900089217, 5.63670654345115775138272135983, 6.85991855225514534749389924063, 8.300776202695755508722174177806, 9.639320798444214170176583959261, 10.06446334829951442712349594126, 10.70706861276775007069914876685, 12.04481107622521975592748609339
