L(s) = 1 | + (−0.494 − 5.17i)3-s + (2.99 − 5.19i)5-s + (−0.375 + 18.5i)7-s + (−26.5 + 5.11i)9-s + (−39.3 + 22.7i)11-s + (−22.7 − 13.1i)13-s + (−28.3 − 12.9i)15-s − 19.7·17-s + 27.9i·19-s + (95.9 − 7.21i)21-s + (−60.3 − 34.8i)23-s + (44.5 + 77.0i)25-s + (39.5 + 134. i)27-s + (−119. + 68.7i)29-s + (138. + 79.7i)31-s + ⋯ |
L(s) = 1 | + (−0.0951 − 0.995i)3-s + (0.268 − 0.464i)5-s + (−0.0202 + 0.999i)7-s + (−0.981 + 0.189i)9-s + (−1.07 + 0.623i)11-s + (−0.485 − 0.280i)13-s + (−0.488 − 0.222i)15-s − 0.281·17-s + 0.337i·19-s + (0.997 − 0.0749i)21-s + (−0.547 − 0.315i)23-s + (0.356 + 0.616i)25-s + (0.282 + 0.959i)27-s + (−0.762 + 0.440i)29-s + (0.800 + 0.462i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4181988089\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4181988089\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.494 + 5.17i)T \) |
| 7 | \( 1 + (0.375 - 18.5i)T \) |
good | 5 | \( 1 + (-2.99 + 5.19i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (39.3 - 22.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (22.7 + 13.1i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 19.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.9iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (60.3 + 34.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (119. - 68.7i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-138. - 79.7i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (20.4 - 35.3i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-55.4 - 95.9i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-109. - 189. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 209. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-413. + 716. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (594. - 343. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (171. - 296. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 387. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 220. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-242. - 419. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (354. + 613. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 140.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.30e3 - 753. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−12.27181977996423584380284384828, −11.10127755525811708413088362439, −9.918871763310917182217228883546, −8.802282357053372122879695111666, −7.994372167829037015874312920690, −6.95726564168597583554838078980, −5.72284486275529207874443799062, −5.00324436088860439462938611980, −2.82515349065035427110942973597, −1.75751364205925980029100254127,
0.15555588416263851197612248315, 2.61746932124063390438523429733, 3.86215689911590056986721522951, 4.96736892297438880423232425803, 6.11081881423716028134786125089, 7.34846432585956255741003623779, 8.459469402837483703422561409991, 9.655911960768378097554101846359, 10.44610998807860854956719073312, 10.94072091361059251446792823963