L(s) = 1 | + 1.73i·3-s + 5-s − 2.44·7-s − 5.65i·13-s + 1.73i·15-s + 4.24i·17-s − 4.24i·21-s + 8.66i·23-s − 4·25-s + 5.19i·27-s + 5.65i·29-s − 1.73i·31-s − 2.44·35-s + 5·37-s + 9.79·39-s + ⋯ |
L(s) = 1 | + 0.999i·3-s + 0.447·5-s − 0.925·7-s − 1.56i·13-s + 0.447i·15-s + 1.02i·17-s − 0.925i·21-s + 1.80i·23-s − 0.800·25-s + 1.00i·27-s + 1.05i·29-s − 0.311i·31-s − 0.414·35-s + 0.821·37-s + 1.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174539299\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174539299\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.66iT - 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 12.1iT - 59T^{2} \) |
| 61 | \( 1 - 5.65iT - 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + 11.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 17T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.784783976769632728784400610501, −8.950826271567072703486394225620, −7.997073213016561572841351722012, −7.19944534676252116390803966165, −6.02646817155231978692560162836, −5.61717676716492713493174662431, −4.60556714529677619921243665335, −3.54777590705541205081973212234, −3.08598317769891658658228517315, −1.48963669398320285573611783192,
0.41620199703343641056279887952, 1.86922909664973591891713306399, 2.55939272453890653065725893124, 3.88113575883600547995745704780, 4.79879379788926519293429243414, 6.00452260883666275322379211693, 6.75235878594781411827518713107, 6.92115656074618073814936712332, 8.063028473535162325115955604248, 8.863255639004802286583158550152