| L(s) = 1 | + 1.23·3-s + 1.23·7-s − 7.47·9-s − 11.4i·11-s − 5.41i·13-s − 6.94i·17-s + 29.8i·19-s + 1.52·21-s + 19.1·23-s − 20.3·27-s + 21.0·29-s − 34.4i·31-s − 14.1i·33-s − 19.3i·37-s − 6.69i·39-s + ⋯ |
| L(s) = 1 | + 0.412·3-s + 0.176·7-s − 0.830·9-s − 1.03i·11-s − 0.416i·13-s − 0.408i·17-s + 1.57i·19-s + 0.0727·21-s + 0.831·23-s − 0.754·27-s + 0.726·29-s − 1.11i·31-s − 0.427i·33-s − 0.521i·37-s − 0.171i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7126172218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7126172218\) |
| \(L(2)\) |
|
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 | \( 1 - 1.23T + 9T^{2} \) |
| 7 | \( 1 - 1.23T + 49T^{2} \) |
| 11 | \( 1 + 11.4iT - 121T^{2} \) |
| 13 | \( 1 + 5.41iT - 169T^{2} \) |
| 17 | \( 1 + 6.94iT - 289T^{2} \) |
| 19 | \( 1 - 29.8iT - 361T^{2} \) |
| 23 | \( 1 - 19.1T + 529T^{2} \) |
| 29 | \( 1 - 21.0T + 841T^{2} \) |
| 31 | \( 1 + 34.4iT - 961T^{2} \) |
| 37 | \( 1 + 19.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 58.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 98.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.19T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.01T + 4.48e3T^{2} \) |
| 71 | \( 1 + 84.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 124. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 160.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 46.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 133. iT - 9.40e3T^{2} \) |
| 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
−8.664212416413323967508405778659, −8.254083768799011439744958855301, −7.49449124308533519388869111486, −6.28610896460701269077989526484, −5.70276630107305793088295291585, −4.75943741235450655639114459033, −3.46043759343597370693417433588, −2.96613848193450833842359303204, −1.61000827254847849828437404066, −0.17194065519187648604018347506,
1.53362668086604598520576605943, 2.61700990131689106823100500549, 3.46619562182995492393333204976, 4.76243407389736469387259061469, 5.18702377736938851369213311239, 6.72650769174636230022790176137, 6.89090167389723491659372142838, 8.307604037454486820094730982078, 8.547533326221601270166664115324, 9.566457509570299008438805026253
