| L(s) = 1 | − i·3-s + 2i·7-s + 2·9-s + 5·11-s + 5i·17-s − 5·19-s + 2·21-s + 6i·23-s − 5i·27-s + 4·29-s − 10·31-s − 5i·33-s + 10i·37-s + 5·41-s − 4i·43-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.755i·7-s + 0.666·9-s + 1.50·11-s + 1.21i·17-s − 1.14·19-s + 0.436·21-s + 1.25i·23-s − 0.962i·27-s + 0.742·29-s − 1.79·31-s − 0.870i·33-s + 1.64i·37-s + 0.780·41-s − 0.609i·43-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
| \(L(1)\) |
\(\approx\) |
\(1.864861475\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864861475\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 5iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.358489985756048941518706710249, −8.670977469165622527725920724351, −7.923179773704908364355655796203, −6.88933176995308127332384693969, −6.38920341941992784175220056308, −5.55457233241086900779968281508, −4.30376623456525155701418241215, −3.59696575161468210870095631863, −2.09809972258832318742786203106, −1.35225898026343701835050404094,
0.812837959421356350827903639239, 2.20180909365265860431114440790, 3.71379414996926114056542173444, 4.17500832262042093160340389994, 4.98587120133807226879397449288, 6.25332556535095987565649285236, 6.95860533905878469011456021695, 7.58813209821588575866016272332, 8.968561195121455879911428114774, 9.165726533024305486658527405683
