L(s) = 1 | + (0.622 − 1.26i)2-s + (−1.22 − 1.58i)4-s − 1.44·7-s + (−2.77 + 0.570i)8-s + 1.14i·11-s + 3.87i·13-s + (−0.902 + 1.84i)14-s + (−0.999 + 3.87i)16-s + 3.05·17-s − 0.710i·19-s + (1.44 + 0.710i)22-s − 5.54·23-s + (4.91 + 2.41i)26-s + (1.77 + 2.29i)28-s + 6.22i·29-s + ⋯ |
L(s) = 1 | + (0.440 − 0.897i)2-s + (−0.612 − 0.790i)4-s − 0.547·7-s + (−0.979 + 0.201i)8-s + 0.344i·11-s + 1.07i·13-s + (−0.241 + 0.491i)14-s + (−0.249 + 0.968i)16-s + 0.739·17-s − 0.163i·19-s + (0.309 + 0.151i)22-s − 1.15·23-s + (0.964 + 0.472i)26-s + (0.335 + 0.433i)28-s + 1.15i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.375662115\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.375662115\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.622 + 1.26i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 - 1.14iT - 11T^{2} \) |
| 13 | \( 1 - 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + 0.710iT - 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 - 6.22iT - 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 6.32iT - 37T^{2} \) |
| 41 | \( 1 - 8.03T + 41T^{2} \) |
| 43 | \( 1 - 7.03iT - 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 - 3.93iT - 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 + 2.45iT - 61T^{2} \) |
| 67 | \( 1 - 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.567683632791142491870298885228, −8.784926232875736763429596181000, −7.80130905305400041019859250826, −6.67218897191792429477532382131, −6.05121409137624518695145245281, −5.00635834431524367503823182252, −4.22691591017394115785743696247, −3.35934275328586753029241658830, −2.37929419790749836720766596997, −1.25566580934427710848524540412,
0.47650774105457580312866090495, 2.57165191350672775133498316551, 3.53400170288551723588177435102, 4.27697993299312862204243688773, 5.55148165713370761486013232354, 5.86353781196128358136355089596, 6.78375833629465176018028995087, 7.79796606279094804035668148252, 8.101295307602043305592547941684, 9.159168577534459363843709643820