L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.608 − 0.793i)3-s + (0.499 + 0.866i)4-s + (−0.130 − 0.991i)6-s + 0.999i·8-s + (−0.258 + 0.965i)9-s + (0.448 + 0.258i)11-s + (0.382 − 0.923i)12-s + (−0.5 + 0.866i)16-s − 0.261·17-s + (−0.707 + 0.707i)18-s + 1.21i·19-s + (0.258 + 0.448i)22-s + (0.793 − 0.608i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.608 − 0.793i)3-s + (0.499 + 0.866i)4-s + (−0.130 − 0.991i)6-s + 0.999i·8-s + (−0.258 + 0.965i)9-s + (0.448 + 0.258i)11-s + (0.382 − 0.923i)12-s + (−0.5 + 0.866i)16-s − 0.261·17-s + (−0.707 + 0.707i)18-s + 1.21i·19-s + (0.258 + 0.448i)22-s + (0.793 − 0.608i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.651106465\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.651106465\) |
\(L(1)\) |
|
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 | \( 1 + (0.608 + 0.793i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 0.261T + T^{2} \) |
| 19 | \( 1 - 1.21iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.793 + 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.58iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 + (-1.71 - 0.991i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.561441179639000933681374879404, −7.84324123980901790555764012315, −7.29468455257523598720359969965, −6.55729264847731575263983337785, −5.91844457446402600477262290491, −5.32769131800438212687247631222, −4.40935877704132894184945400678, −3.60413677162114432921493426430, −2.44935418868777478531954607992, −1.52148995734800611294121360585,
0.806895941818325548827357135264, 2.31076065634306174901957810517, 3.23124673638630604726993368348, 4.13746799787339816671148761662, 4.61726608430342488731325078144, 5.46977134059834355132715305143, 6.18831303626625527541751931173, 6.72020169953726013940087383748, 7.74050231974693315020713797057, 9.100060770893776527377636549956