L(s) = 1 | + (1.5 + 0.866i)3-s − 5-s + (−2 − 1.73i)7-s + (1.5 + 2.59i)9-s + 5.19i·11-s − 1.73i·13-s + (−1.5 − 0.866i)15-s + 3·17-s + 3.46i·19-s + (−1.50 − 4.33i)21-s + 25-s + 5.19i·27-s − 5.19i·29-s + 10.3i·31-s + (−4.5 + 7.79i)33-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s − 0.447·5-s + (−0.755 − 0.654i)7-s + (0.5 + 0.866i)9-s + 1.56i·11-s − 0.480i·13-s + (−0.387 − 0.223i)15-s + 0.727·17-s + 0.794i·19-s + (−0.327 − 0.944i)21-s + 0.200·25-s + 0.999i·27-s − 0.964i·29-s + 1.86i·31-s + (−0.783 + 1.35i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.581032280\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.581032280\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 11 | \( 1 - 5.19iT - 11T^{2} \) |
| 13 | \( 1 + 1.73iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 5.19iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.73iT - 97T^{2} \) |
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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
−9.733749040158929515548400794686, −8.851558402607513733528858424549, −7.911767255622625375704060215739, −7.40190163913803069998591671638, −6.63585771449316877949278701974, −5.29591712777220062768202091441, −4.41701088323047055941457096225, −3.66216343744557861561566676306, −2.88663872646298472636726172871, −1.55202335156257597807165764069,
0.53656611710741931084683969148, 2.08476272030609904313013749633, 3.26450710803956470259550416904, 3.54415772029358230774224103448, 5.01702267698658122950743633720, 6.12237931287808580117975900809, 6.67623566099226817054563724135, 7.68414124888335670709591020634, 8.356976894471798705370994653717, 9.050455057028632390964115449926