L(s) = 1 | − 2-s + (−0.396 − 1.68i)3-s + 4-s − 2.37i·5-s + (0.396 + 1.68i)6-s + (0.792 − 2.52i)7-s − 8-s + (−2.68 + 1.33i)9-s + 2.37i·10-s − 5.74·11-s + (−0.396 − 1.68i)12-s + (−3.46 − i)13-s + (−0.792 + 2.52i)14-s + (−4 + 0.939i)15-s + 16-s + 2.52·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.228 − 0.973i)3-s + 0.5·4-s − 1.06i·5-s + (0.161 + 0.688i)6-s + (0.299 − 0.954i)7-s − 0.353·8-s + (−0.895 + 0.445i)9-s + 0.750i·10-s − 1.73·11-s + (−0.114 − 0.486i)12-s + (−0.960 − 0.277i)13-s + (−0.211 + 0.674i)14-s + (−1.03 + 0.242i)15-s + 0.250·16-s + 0.612·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0924397 + 0.516195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0924397 + 0.516195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.396 + 1.68i)T \) |
| 7 | \( 1 + (-0.792 + 2.52i)T \) |
| 13 | \( 1 + (3.46 + i)T \) |
good | 5 | \( 1 + 2.37iT - 5T^{2} \) |
| 11 | \( 1 + 5.74T + 11T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 7.57T + 19T^{2} \) |
| 23 | \( 1 - 6.78iT - 23T^{2} \) |
| 29 | \( 1 + 7.57iT - 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 - 4.90iT - 37T^{2} \) |
| 41 | \( 1 + 2.62iT - 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 4.62iT - 47T^{2} \) |
| 53 | \( 1 + 3.16iT - 53T^{2} \) |
| 59 | \( 1 + 8.74iT - 59T^{2} \) |
| 61 | \( 1 + 3.74iT - 61T^{2} \) |
| 67 | \( 1 - 2.67iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 + 9.37T + 79T^{2} \) |
| 83 | \( 1 + 4.74iT - 83T^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + 7.42T + 97T^{2} \) |
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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
−10.16875184344090145437834065535, −9.526531399659650479470586404378, −8.102636121557006808030046283622, −7.78413650873342870090744619652, −7.13804705526074775997474143145, −5.50759637044448920748166450221, −5.06557860570888441964676601138, −3.08087825152918186629949200606, −1.59128937374608044902468963796, −0.38195970427260985084010570015,
2.53399417894280992286062283581, 3.15774551103178879568000635308, 5.05539456520460790860644034284, 5.57227249256244313902710366992, 6.93855176382118367363246366784, 7.80109299353199650773114610440, 8.794246164518138413841132014602, 9.708182492371031307792793846814, 10.39312085686292696113266785616, 10.95028297862443995244301327074