L(s) = 1 | + (0.766 + 0.642i)2-s + (0.972 − 0.816i)3-s + (0.173 + 0.984i)4-s + (−1.43 − 0.524i)5-s + 1.27·6-s + (−1.82 − 0.665i)7-s + (−0.500 + 0.866i)8-s + (−0.240 + 1.36i)9-s + (−0.766 − 1.32i)10-s + (−0.220 + 0.381i)11-s + (0.972 + 0.816i)12-s + (−0.367 − 2.08i)13-s + (−0.972 − 1.68i)14-s + (−1.82 + 0.665i)15-s + (−0.939 + 0.342i)16-s + (−0.664 + 3.76i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (0.561 − 0.471i)3-s + (0.0868 + 0.492i)4-s + (−0.643 − 0.234i)5-s + 0.518·6-s + (−0.691 − 0.251i)7-s + (−0.176 + 0.306i)8-s + (−0.0802 + 0.455i)9-s + (−0.242 − 0.419i)10-s + (−0.0664 + 0.115i)11-s + (0.280 + 0.235i)12-s + (−0.101 − 0.578i)13-s + (−0.260 − 0.450i)14-s + (−0.472 + 0.171i)15-s + (−0.234 + 0.0855i)16-s + (−0.161 + 0.913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19885 + 0.156430i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19885 + 0.156430i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (-2.18 - 5.67i)T \) |
good | 3 | \( 1 + (-0.972 + 0.816i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (1.43 + 0.524i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.82 + 0.665i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (0.220 - 0.381i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.367 + 2.08i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.664 - 3.76i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-4.55 + 3.82i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (2.85 + 4.94i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.36 - 2.36i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 41 | \( 1 + (-1.80 - 10.2i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 + (3.84 + 6.65i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.62 - 0.591i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.93 - 0.702i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.89 + 10.7i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.57 - 0.572i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (7.70 - 6.46i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 + (12.5 + 4.56i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.617 + 3.50i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-11.1 + 4.05i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.296 - 0.513i)T + (-48.5 + 84.0i)T^{2} \) |
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
−14.56068789180220549373427425034, −13.45893230502338783113568116388, −12.82734882210553702249328146600, −11.65408780270337539405736842590, −10.13472310669363245309328951137, −8.454404909346440631883384600082, −7.68086500005625822024114520606, −6.39590487105428743063589574086, −4.64619723666353755724299713111, −3.00192275323802422962134276611,
3.02637810980931561368589511780, 4.10194151101519860627686266291, 5.94802992945798473694835313713, 7.52473350314672694987007075073, 9.199619352110332005793130174317, 9.940305223876038655807837984685, 11.53383375375361415502524236460, 12.16741953995365953990220538600, 13.64189587310173813630305192099, 14.41065871790534941523011481958