| L(s) = 1 | − 1.41·2-s + (2.98 + 0.318i)3-s + 2.00·4-s + (−4.21 − 0.450i)6-s + 2.64i·7-s − 2.82·8-s + (8.79 + 1.89i)9-s + 2.46i·11-s + (5.96 + 0.636i)12-s + 5.95i·13-s − 3.74i·14-s + 4.00·16-s + 17.1·17-s + (−12.4 − 2.68i)18-s + 10.3·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.994 + 0.106i)3-s + 0.500·4-s + (−0.703 − 0.0750i)6-s + 0.377i·7-s − 0.353·8-s + (0.977 + 0.211i)9-s + 0.223i·11-s + (0.497 + 0.0530i)12-s + 0.458i·13-s − 0.267i·14-s + 0.250·16-s + 1.00·17-s + (−0.691 − 0.149i)18-s + 0.545·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.031673885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.031673885\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.98 - 0.318i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 2.46iT - 121T^{2} \) |
| 13 | \( 1 - 5.95iT - 169T^{2} \) |
| 17 | \( 1 - 17.1T + 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 + 25.4T + 529T^{2} \) |
| 29 | \( 1 + 2.22iT - 841T^{2} \) |
| 31 | \( 1 - 8.60T + 961T^{2} \) |
| 37 | \( 1 - 24.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 10.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 63.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 64.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 25.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 43.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 96.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 10.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 161. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 167. iT - 9.40e3T^{2} \) |
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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
−9.774661339180747723287899964332, −9.061695282197386467650134910660, −8.201686613387301847091053305469, −7.66632510711926477882941026175, −6.74909943056252770528621189806, −5.68589518725014348353498133906, −4.43821262180736262271618657626, −3.34460197687737387130299528039, −2.37528846923633849901214927555, −1.28673490564940848223213227375,
0.76621457724641804843643703915, 1.98535425805884101327900649161, 3.13985820189397812803683240409, 3.94602046026074902529130456780, 5.35866948567263862211169445711, 6.45733466952478726977365833894, 7.48631122597450545792330951648, 7.904508446885024267697248115990, 8.723016595751363785589577539628, 9.592810534275825849518335434042
