| L(s) = 1 | − 2.32i·5-s − 2.64·7-s − 0.412i·11-s − 4.58·13-s + 32.4i·17-s + 20.4·19-s − 35.1i·23-s + 19.5·25-s − 25.0i·29-s + 43.2·31-s + 6.15i·35-s − 21.4·37-s − 59.5i·41-s − 51.7·43-s − 5.47i·47-s + ⋯ |
| L(s) = 1 | − 0.465i·5-s − 0.377·7-s − 0.0374i·11-s − 0.352·13-s + 1.90i·17-s + 1.07·19-s − 1.52i·23-s + 0.783·25-s − 0.863i·29-s + 1.39·31-s + 0.175i·35-s − 0.578·37-s − 1.45i·41-s − 1.20·43-s − 0.116i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.655562494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655562494\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 2.32iT - 25T^{2} \) |
| 11 | \( 1 + 0.412iT - 121T^{2} \) |
| 13 | \( 1 + 4.58T + 169T^{2} \) |
| 17 | \( 1 - 32.4iT - 289T^{2} \) |
| 19 | \( 1 - 20.4T + 361T^{2} \) |
| 23 | \( 1 + 35.1iT - 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 - 43.2T + 961T^{2} \) |
| 37 | \( 1 + 21.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 59.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5.47iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 40.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 88.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 80.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 24.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 33.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22T + 6.24e3T^{2} \) |
| 83 | \( 1 + 30.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 73.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 34.7T + 9.40e3T^{2} \) |
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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
−9.818601353291520603882000998168, −8.558550305430143741471629520418, −8.300207980081192511381056420271, −7.01130570301978794651352115135, −6.27459742526365807765098895960, −5.29031259231644743768664186478, −4.34572503204871134625135751154, −3.33193088080871465899620881577, −2.05720592061107714346692608764, −0.62974104681610449471974648052,
1.05474174890426694192742606654, 2.72853192057293909084804595461, 3.34046681368507292961770438310, 4.79533844233064028416943745991, 5.47443214851210211555782185700, 6.76782633757088581149177939078, 7.19428677971233183911206471587, 8.154311927722379435554590263047, 9.370236200398006157979640759880, 9.705037307871854275108922703710
