L(s) = 1 | + 1.41i·2-s + (0.821 + 2.88i)3-s − 2.00·4-s + (−4.08 + 1.16i)6-s − 2.64·7-s − 2.82i·8-s + (−7.64 + 4.74i)9-s − 18.1i·11-s + (−1.64 − 5.77i)12-s − 22.5·13-s − 3.74i·14-s + 4.00·16-s − 0.457i·17-s + (−6.70 − 10.8i)18-s + 30.1·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.273 + 0.961i)3-s − 0.500·4-s + (−0.680 + 0.193i)6-s − 0.377·7-s − 0.353i·8-s + (−0.849 + 0.526i)9-s − 1.65i·11-s + (−0.136 − 0.480i)12-s − 1.73·13-s − 0.267i·14-s + 0.250·16-s − 0.0268i·17-s + (−0.372 − 0.600i)18-s + 1.58·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.283096330\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.283096330\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-0.821 - 2.88i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 11 | \( 1 + 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 22.5T + 169T^{2} \) |
| 17 | \( 1 + 0.457iT - 289T^{2} \) |
| 19 | \( 1 - 30.1T + 361T^{2} \) |
| 23 | \( 1 - 12.3iT - 529T^{2} \) |
| 29 | \( 1 + 4.70iT - 841T^{2} \) |
| 31 | \( 1 - 45.2T + 961T^{2} \) |
| 37 | \( 1 - 32.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 22.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 20.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.67iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.97iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 67.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 20.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 121. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 151.T + 9.40e3T^{2} \) |
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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
−9.657795080449413573592018569284, −9.014669876910290782668989434852, −8.074852859250884056287504307106, −7.42350335188752649211858598305, −6.19427422677398845519027228559, −5.42018844310077813020022958413, −4.69234201120329459460195283763, −3.49309649869980709547825907117, −2.76914315429404488556098715594, −0.44978232769951710181293770380,
1.04540767449497491910220477448, 2.33023717752457179723584419005, 2.89744908167777375434582380131, 4.37802996229680813424419828110, 5.21418370269992922883345014293, 6.47667498036846029070964682199, 7.39615602928090972294347522012, 7.77988458727229232709816548303, 9.110465168562684247162350100335, 9.706553779375002014474041953969