L(s) = 1 | + (0.292 − 1.70i)3-s + (0.707 − 2.12i)5-s + (1 + i)7-s + (−2.82 − i)9-s − 1.41i·11-s + (−3.41 − 1.82i)15-s + (−1.41 + 1.41i)17-s − 4i·19-s + (2 − 1.41i)21-s + (2.82 + 2.82i)23-s + (−3.99 − 3i)25-s + (−2.53 + 4.53i)27-s + 7.07·29-s + 2·31-s + (−2.41 − 0.414i)33-s + ⋯ |
L(s) = 1 | + (0.169 − 0.985i)3-s + (0.316 − 0.948i)5-s + (0.377 + 0.377i)7-s + (−0.942 − 0.333i)9-s − 0.426i·11-s + (−0.881 − 0.472i)15-s + (−0.342 + 0.342i)17-s − 0.917i·19-s + (0.436 − 0.308i)21-s + (0.589 + 0.589i)23-s + (−0.799 − 0.600i)25-s + (−0.487 + 0.872i)27-s + 1.31·29-s + 0.359·31-s + (−0.420 − 0.0721i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.978158 - 0.919405i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.978158 - 0.919405i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 + 1.70i)T \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
good | 7 | \( 1 + (-1 - i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (-6 - 6i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-2.82 - 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−11.91500856088757303914227764973, −11.32549753105443260022881061788, −9.777768735286923549198064032291, −8.656557568382396809254784069579, −8.220307151502032302784688510121, −6.84125817143758795961452044454, −5.80466371356607943799907301478, −4.69688962225048418572061635101, −2.74865019178506398584206151122, −1.22294184313253671895092966857,
2.48219697651783341661105068535, 3.80313014105359759317226301696, 4.95608830693979098888685878477, 6.23713535916788691914735255083, 7.41233023072329660843655268632, 8.567044678880102270845229872779, 9.709039525516812508318820575832, 10.43511153928073453256784670117, 11.09182980429537591883014956127, 12.16447141339352205660376059511