L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s + 1.00i·4-s + (1 + 2i)5-s + 1.00i·6-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−0.707 + 2.12i)10-s − 4.34·11-s + (−0.707 + 0.707i)12-s + (−1.93 − 1.93i)13-s + (−0.707 + 2.12i)15-s − 1.00·16-s + (−1.07 + 1.07i)17-s + (−0.707 + 0.707i)18-s − 0.585·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (0.447 + 0.894i)5-s + 0.408i·6-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.223 + 0.670i)10-s − 1.30·11-s + (−0.204 + 0.204i)12-s + (−0.535 − 0.535i)13-s + (−0.182 + 0.547i)15-s − 0.250·16-s + (−0.259 + 0.259i)17-s + (−0.166 + 0.166i)18-s − 0.134·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.895673448\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.895673448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 + (1.93 + 1.93i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.07 - 1.07i)T - 17iT^{2} \) |
| 19 | \( 1 + 0.585T + 19T^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 23iT^{2} \) |
| 29 | \( 1 - 9.90iT - 29T^{2} \) |
| 31 | \( 1 - 3.65iT - 31T^{2} \) |
| 37 | \( 1 + (-3.27 - 3.27i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.03iT - 41T^{2} \) |
| 43 | \( 1 + (-6.17 + 6.17i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.68 + 5.68i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.45 - 9.45i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.55T + 59T^{2} \) |
| 61 | \( 1 + 8.68iT - 61T^{2} \) |
| 67 | \( 1 + (-5.93 - 5.93i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + (-7.65 - 7.65i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.03iT - 79T^{2} \) |
| 83 | \( 1 + (1.61 + 1.61i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.00T + 89T^{2} \) |
| 97 | \( 1 + (-9.85 + 9.85i)T - 97iT^{2} \) |
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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
−9.993792758896051214175379319815, −9.037773447958863487190398368784, −8.112560275217115002446327540654, −7.42882662503556310395380943135, −6.70879652141303694131097868989, −5.56334225220621246452191264828, −5.14386203998017206224302888201, −3.85589175717305615286252551007, −2.99445909610562537371926986892, −2.20494241279844083714524479867,
0.55298089326206392523930698340, 2.11019600324419071788521878592, 2.59189691709226788392345526748, 4.12090980788416922795671827225, 4.77663671249819919141439401070, 5.73822750251342500305763108342, 6.46818771166479002067640354142, 7.76838214391109877809831569249, 8.192590446686169804033535841865, 9.425881597081021298734961112317