| L(s) = 1 | − 2.23i·3-s − 4.47i·7-s − 2.00·9-s − 2.23·11-s − 4i·13-s + 7i·17-s − 6.70·19-s − 10.0·21-s − 4.47i·23-s − 2.23i·27-s + 4.47·31-s + 5.00i·33-s + 2i·37-s − 8.94·39-s + 5·41-s + ⋯ |
| L(s) = 1 | − 1.29i·3-s − 1.69i·7-s − 0.666·9-s − 0.674·11-s − 1.10i·13-s + 1.69i·17-s − 1.53·19-s − 2.18·21-s − 0.932i·23-s − 0.430i·27-s + 0.803·31-s + 0.870i·33-s + 0.328i·37-s − 1.43·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.016971284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.016971284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 2.23iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9iT - 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.506168062339636714672037169895, −7.992757936050774065659583855319, −7.47907843996653230068374465684, −6.50243333512293403461219937792, −6.11321347121725484928627928177, −4.66440526546833807150705917231, −3.86385685788537873863377961264, −2.60154891771284060930980880101, −1.43934083527078678654758243277, −0.38791313659385024695453233007,
2.15805784182267854612970113496, 2.92597694131702203629127479998, 4.14729472541376977666057808528, 4.92032605350189200925364142557, 5.53658995140749345259213975322, 6.47307415905384557449710263302, 7.55350663774047314846543323583, 8.760902735371488909899955177382, 9.031594695143234072550802326252, 9.736058462645231798819897948367
