| L(s) = 1 | − 1.41·2-s + 2.00·4-s + 2.64i·7-s − 2.82·8-s + 5.15i·11-s − 6.04i·13-s − 3.74i·14-s + 4.00·16-s + 5.99·17-s + 10.0·19-s − 7.29i·22-s − 20.1·23-s + 8.54i·26-s + 5.29i·28-s − 26.5i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.377i·7-s − 0.353·8-s + 0.468i·11-s − 0.464i·13-s − 0.267i·14-s + 0.250·16-s + 0.352·17-s + 0.529·19-s − 0.331i·22-s − 0.874·23-s + 0.328i·26-s + 0.188i·28-s − 0.915i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7713401869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7713401869\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 5.15iT - 121T^{2} \) |
| 13 | \( 1 + 6.04iT - 169T^{2} \) |
| 17 | \( 1 - 5.99T + 289T^{2} \) |
| 19 | \( 1 - 10.0T + 361T^{2} \) |
| 23 | \( 1 + 20.1T + 529T^{2} \) |
| 29 | \( 1 + 26.5iT - 841T^{2} \) |
| 31 | \( 1 + 6.40T + 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.37iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 48.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 35.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 85.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 42.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 35.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 49.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 17.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 123. iT - 9.40e3T^{2} \) |
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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
−8.206463823212787550029124230392, −7.76068083457892994318847842697, −6.93785846541331414569595931758, −6.07253529206047017883214137055, −5.42675759178373387587164052588, −4.40581767302322388974132828452, −3.36054574324800929723996175271, −2.44324445303183743667783402439, −1.49872002384531700481362499020, −0.24940592664430470348657658008,
0.969011929952314174790578451572, 1.92676391482890620773024493255, 3.08617759108165554548770743204, 3.86020630365891164023409034488, 4.94432665807180790357194990184, 5.82075226947935105512452443961, 6.60684102409870530688292821288, 7.32964009870104707247754876404, 8.000582306463805866361055553901, 8.726176773518377913694685074904
