| L(s) = 1 | − 2.03i·3-s − 2.23·5-s + 4.27·7-s + 4.86·9-s + 5.47·11-s − 0.634i·13-s + 4.54i·15-s − 1.69·17-s + (−6.46 − 17.8i)19-s − 8.70i·21-s + 25.9·23-s + 5.00·25-s − 28.1i·27-s − 48.8i·29-s − 4.47i·31-s + ⋯ |
| L(s) = 1 | − 0.677i·3-s − 0.447·5-s + 0.611·7-s + 0.540·9-s + 0.497·11-s − 0.0487i·13-s + 0.303i·15-s − 0.0995·17-s + (−0.340 − 0.940i)19-s − 0.414i·21-s + 1.12·23-s + 0.200·25-s − 1.04i·27-s − 1.68i·29-s − 0.144i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41620 - 0.993777i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41620 - 0.993777i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 19 | \( 1 + (6.46 + 17.8i)T \) |
| good | 3 | \( 1 + 2.03iT - 9T^{2} \) |
| 7 | \( 1 - 4.27T + 49T^{2} \) |
| 11 | \( 1 - 5.47T + 121T^{2} \) |
| 13 | \( 1 + 0.634iT - 169T^{2} \) |
| 17 | \( 1 + 1.69T + 289T^{2} \) |
| 23 | \( 1 - 25.9T + 529T^{2} \) |
| 29 | \( 1 + 48.8iT - 841T^{2} \) |
| 31 | \( 1 + 4.47iT - 961T^{2} \) |
| 37 | \( 1 - 6.12iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 16.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 0.690T + 1.84e3T^{2} \) |
| 47 | \( 1 - 23.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 0.251iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 39.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 96.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 74.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 1.29T + 6.88e3T^{2} \) |
| 89 | \( 1 + 141. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 125. iT - 9.40e3T^{2} \) |
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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
−11.23562283611005334150645089735, −10.07976649951113533412241936564, −8.983896471800554347464606239102, −8.046858880764062467972152915782, −7.20456493626912154752178256298, −6.41436708281756989039908930781, −4.96565690909153954750977957639, −3.97391314131446325663313583482, −2.32461347737730889218970131506, −0.889868202891718338234430088653,
1.46531709373388902404073703841, 3.36147588083395745574807349395, 4.36248198920234725526400925072, 5.21405105166450097218812717916, 6.65424643343714789984079237559, 7.60222309516453395891563968078, 8.665799099493095779563285861971, 9.462976098970562164427372501185, 10.56342917789053456425133756102, 11.09352394787812291275641746999
