L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s − 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s − 0.999·20-s − 0.517·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.707 − 0.707i)7-s − 0.999i·8-s + (0.866 − 0.499i)10-s + (0.448 − 0.258i)11-s − 1.93i·13-s + (−0.965 + 0.258i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)19-s − 0.999·20-s − 0.517·22-s + (−1.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.965 + 1.67i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6142357908\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6142357908\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + 1.93iT - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - 0.517T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.620122807928384260136288486336, −8.149705917205951800720585116256, −7.61468838232533713679516408383, −6.78540233262014006190101645260, −6.05945313882672549321221121267, −4.66665754908628450691000349812, −3.75742697709939887861895654743, −3.04963537027478760152322040675, −2.00775101375048727644421955613, −0.53703348900339963145703615923,
1.58154652192447398187998656655, 2.07566503720676041327356890164, 4.01708366109892423749793215733, 4.58971482727373633547565068252, 5.63711456712415529308045863432, 6.24780050485837676856553394770, 7.27831442044283802170128861062, 7.87747801511242124519473351291, 8.666614790443279086323105697239, 9.149022989276011331648470699777