L(s) = 1 | − 2.61·5-s + (−1.25 + 2.32i)7-s − 4.29i·11-s + 1.53i·13-s − 0.448·17-s − 1.92i·19-s + 0.737i·23-s + 1.82·25-s − 1.41i·29-s + 6.58i·31-s + (3.29 − 6.08i)35-s + 2.82·37-s − 6.94·41-s − 9.64·43-s − 2.72·47-s + ⋯ |
L(s) = 1 | − 1.16·5-s + (−0.475 + 0.879i)7-s − 1.29i·11-s + 0.424i·13-s − 0.108·17-s − 0.442i·19-s + 0.153i·23-s + 0.365·25-s − 0.262i·29-s + 1.18i·31-s + (0.556 − 1.02i)35-s + 0.464·37-s − 1.08·41-s − 1.47·43-s − 0.397·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9983771118\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9983771118\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.25 - 2.32i)T \) |
good | 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 4.29iT - 11T^{2} \) |
| 13 | \( 1 - 1.53iT - 13T^{2} \) |
| 17 | \( 1 + 0.448T + 17T^{2} \) |
| 19 | \( 1 + 1.92iT - 19T^{2} \) |
| 23 | \( 1 - 0.737iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.58iT - 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 + 9.64T + 43T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 - 9.30T + 59T^{2} \) |
| 61 | \( 1 + 8.28iT - 61T^{2} \) |
| 67 | \( 1 - 1.47T + 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 - 11.0iT - 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 6.75iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.359478270210655103638845117238, −7.986437194194066677955168540748, −6.83666620667875341573150762471, −6.42209880457190340871234307946, −5.41632515420390852473945115847, −4.74933972432832894341243844509, −3.54044779135271894369507224275, −3.29427262641281162575675100181, −2.07720715869406953063428560199, −0.55582249469409454705701141952,
0.55607910348440635149366897137, 1.91481825195128275360176116788, 3.15525042270086619980322527103, 3.90726469912225245952240002085, 4.44748205596655971037582136506, 5.31186241986088279898935417274, 6.49734125003070639266446140894, 7.01467752200907610600054076710, 7.79251092496350566064608018415, 8.093334983279838631486189243253