| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (−4.59 − 1.96i)5-s + (6.81 − 1.57i)7-s + 2.82i·8-s + (−2.77 + 6.50i)10-s + 8.15·11-s + 14.6·13-s + (−2.23 − 9.64i)14-s + 4.00·16-s − 5.81·17-s + 33.7i·19-s + (9.19 + 3.93i)20-s − 11.5i·22-s − 37.2i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (−0.919 − 0.393i)5-s + (0.974 − 0.225i)7-s + 0.353i·8-s + (−0.277 + 0.650i)10-s + 0.741·11-s + 1.12·13-s + (−0.159 − 0.688i)14-s + 0.250·16-s − 0.342·17-s + 1.77i·19-s + (0.459 + 0.196i)20-s − 0.524i·22-s − 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.643373187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643373187\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.59 + 1.96i)T \) |
| 7 | \( 1 + (-6.81 + 1.57i)T \) |
| good | 11 | \( 1 - 8.15T + 121T^{2} \) |
| 13 | \( 1 - 14.6T + 169T^{2} \) |
| 17 | \( 1 + 5.81T + 289T^{2} \) |
| 19 | \( 1 - 33.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2iT - 529T^{2} \) |
| 29 | \( 1 - 9.25T + 841T^{2} \) |
| 31 | \( 1 + 19.2iT - 961T^{2} \) |
| 37 | \( 1 + 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 23.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.99iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 82.7T + 9.40e3T^{2} \) |
| 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
−10.49332678580191293712154023199, −9.172708103919085886940106458080, −8.399201635497224411093489907186, −7.88785315265219643832168380146, −6.56349246894212484882965719240, −5.32682072106640718408841446940, −4.14149034343580666291615390295, −3.75568791409627722761350542283, −1.94087984889607727135048288076, −0.75828444999571977052619155147,
1.20686226410569215620850276640, 3.13175760489319499717358857388, 4.25467550937040520258398444066, 5.06223779512397556159637784750, 6.34903758122139736171807571033, 7.07008561589692863209265598934, 8.023895720410066580226137225315, 8.649290103897536755021309049299, 9.488594774197927477774077125645, 10.95054465922453733371843332239
