| L(s) = 1 | + 1.14i·3-s + 3.83i·5-s + 7-s + 1.68·9-s − 4.68i·11-s + 5.53i·13-s − 4.39·15-s + 0.292·17-s + 5.14i·19-s + 1.14i·21-s + 4.97·23-s − 9.68·25-s + 5.37i·27-s + 4.29i·29-s + 7.66·31-s + ⋯ |
| L(s) = 1 | + 0.661i·3-s + 1.71i·5-s + 0.377·7-s + 0.561·9-s − 1.41i·11-s + 1.53i·13-s − 1.13·15-s + 0.0709·17-s + 1.18i·19-s + 0.250i·21-s + 1.03·23-s − 1.93·25-s + 1.03i·27-s + 0.797i·29-s + 1.37·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.845175668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.845175668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.14iT - 3T^{2} \) |
| 5 | \( 1 - 3.83iT - 5T^{2} \) |
| 11 | \( 1 + 4.68iT - 11T^{2} \) |
| 13 | \( 1 - 5.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.292T + 17T^{2} \) |
| 19 | \( 1 - 5.14iT - 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 - 4.29iT - 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 + 9.66iT - 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 5.27iT - 43T^{2} \) |
| 47 | \( 1 + 2.29T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 9.93iT - 59T^{2} \) |
| 61 | \( 1 - 4.16iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.37T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 4.81iT - 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.698968953516536092642712665508, −8.902467911059987541390136329272, −7.959453138241997040765518661456, −7.04195623845952458735483104248, −6.51174525680870965976393860463, −5.63802900558039296742904752235, −4.45564025527574278606092520639, −3.63843455905308069567328940906, −2.94342272301631178651416677331, −1.63892649321759467651755725620,
0.74564730519359422836459060412, 1.52087772071655196221635976317, 2.72620693710905562439493956044, 4.39474128722202801232121293555, 4.78009628614143026399284287745, 5.55968672877126115927025749275, 6.76951042067222193430995405418, 7.50054542878987450500709563872, 8.214030352295197677231767173493, 8.789502369321296790910623369657
