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 | \( 1 + (-0.296 + 0.513i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.41065871790534941523011481958, −13.64189587310173813630305192099, −12.16741953995365953990220538600, −11.53383375375361415502524236460, −9.940305223876038655807837984685, −9.199619352110332005793130174317, −7.52473350314672694987007075073, −5.94802992945798473694835313713, −4.10194151101519860627686266291, −3.02637810980931561368589511780,
3.00192275323802422962134276611, 4.64619723666353755724299713111, 6.39590487105428743063589574086, 7.68086500005625822024114520606, 8.454404909346440631883384600082, 10.13472310669363245309328951137, 11.65408780270337539405736842590, 12.82734882210553702249328146600, 13.45893230502338783113568116388, 14.56068789180220549373427425034