L(s) = 1 | + (−1.40 − 0.192i)2-s + (1.92 + 0.540i)4-s − 0.0802·7-s + (−2.59 − 1.12i)8-s − 2.41i·11-s + 5.26i·13-s + (0.112 + 0.0154i)14-s + (3.41 + 2.08i)16-s − 0.255·17-s − 6.95i·19-s + (−0.465 + 3.38i)22-s − 1.64·23-s + (1.01 − 7.38i)26-s + (−0.154 − 0.0433i)28-s + 4.51i·29-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.136i)2-s + (0.962 + 0.270i)4-s − 0.0303·7-s + (−0.917 − 0.398i)8-s − 0.728i·11-s + 1.46i·13-s + (0.0300 + 0.00413i)14-s + (0.854 + 0.520i)16-s − 0.0620·17-s − 1.59i·19-s + (−0.0993 + 0.721i)22-s − 0.343·23-s + (0.199 − 1.44i)26-s + (−0.0292 − 0.00819i)28-s + 0.838i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038524722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038524722\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 + 0.192i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 2.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.26iT - 13T^{2} \) |
| 17 | \( 1 + 0.255T + 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 - 2.67iT - 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4.08iT - 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 7.27iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 9.20iT - 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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
−9.207879294246826009784068501772, −8.584268055614635136005511535113, −7.82182983308197088975333097776, −6.70801088800475187313756793340, −6.53989934217533913833350795806, −5.22286475079971363246228640879, −4.11625340654151903152136972298, −2.99546464555679630585753658182, −2.05985667270012139230754413753, −0.73577810483741936086093001696,
0.884012783868078361687189348819, 2.15790180882278036575941617438, 3.13527494556327164187592507861, 4.35785793050621991164474396894, 5.64787770589571723464644157284, 6.12142256911452880723911306712, 7.19799768400644860600469188971, 7.985945029798666842675010511823, 8.271553524849644415520035898285, 9.524795656942592085785676648595