| L(s) = 1 | + 1.41·2-s + 2.00·4-s − 2.64i·7-s + 2.82·8-s − 3.78i·11-s − 14.9i·13-s − 3.74i·14-s + 4.00·16-s + 0.335·17-s + 29.8·19-s − 5.35i·22-s − 18.5·23-s − 21.1i·26-s − 5.29i·28-s − 11.1i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s − 0.377i·7-s + 0.353·8-s − 0.344i·11-s − 1.14i·13-s − 0.267i·14-s + 0.250·16-s + 0.0197·17-s + 1.57·19-s − 0.243i·22-s − 0.805·23-s − 0.813i·26-s − 0.188i·28-s − 0.384i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.868458991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868458991\) |
| \(L(2)\) |
|
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 | \( 1 + 3.78iT - 121T^{2} \) |
| 13 | \( 1 + 14.9iT - 169T^{2} \) |
| 17 | \( 1 - 0.335T + 289T^{2} \) |
| 19 | \( 1 - 29.8T + 361T^{2} \) |
| 23 | \( 1 + 18.5T + 529T^{2} \) |
| 29 | \( 1 + 11.1iT - 841T^{2} \) |
| 31 | \( 1 - 0.603T + 961T^{2} \) |
| 37 | \( 1 - 41.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 50.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 44.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 102.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 17.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 76.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 60.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 129.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 15.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 59.4iT - 9.40e3T^{2} \) |
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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
−8.013305898882126176171345965172, −7.61517315536220190067660464910, −6.71780259990281451745029682216, −5.85176930501671374487490700864, −5.30265456163524093124161231256, −4.42963887106689911280731269425, −3.44448792175189618856422834073, −2.92988851114589160807917495417, −1.62540055180879851041919727203, −0.48020371113298241678087542965,
1.31075174285851434439018591641, 2.24774047887963360623658420998, 3.20567226507382102547156060040, 4.08409701672338147952058690331, 4.85475520918235498853967549032, 5.61406875171988782746670059651, 6.36967903012344984831751752170, 7.14556200310677807241263319948, 7.77794930518502379027636109902, 8.729876041046859715634930369121
