L(s) = 1 | + (1.68 + 0.420i)3-s + (−1.08 + 1.95i)5-s + (−0.595 + 2.57i)7-s + (2.64 + 1.41i)9-s + 2.82i·11-s + 3.36·13-s + (−2.64 + 2.82i)15-s − 4.75i·17-s + 5.59i·19-s + (−2.08 + 4.08i)21-s − 7.29·23-s + (−2.64 − 4.24i)25-s + (3.85 + 3.48i)27-s + 0.500i·29-s − 3.06i·31-s + ⋯ |
L(s) = 1 | + (0.970 + 0.242i)3-s + (−0.485 + 0.874i)5-s + (−0.224 + 0.974i)7-s + (0.881 + 0.471i)9-s + 0.852i·11-s + 0.931·13-s + (−0.683 + 0.730i)15-s − 1.15i·17-s + 1.28i·19-s + (−0.454 + 0.890i)21-s − 1.52·23-s + (−0.529 − 0.848i)25-s + (0.740 + 0.671i)27-s + 0.0930i·29-s − 0.551i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.979277484\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.979277484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 - 0.420i)T \) |
| 5 | \( 1 + (1.08 - 1.95i)T \) |
| 7 | \( 1 + (0.595 - 2.57i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 - 0.500iT - 29T^{2} \) |
| 31 | \( 1 + 3.06iT - 31T^{2} \) |
| 37 | \( 1 - 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 7.82iT - 47T^{2} \) |
| 53 | \( 1 + 8.58T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 2.52iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 9.81iT - 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 + 6.97iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−9.720124202533982502562484930129, −8.785300676065454103557151069835, −8.035154337830980377380875960115, −7.44760797502299948575254776804, −6.50028934624703084326461285366, −5.64571411427149646699087982462, −4.36271215131558410835300507499, −3.63028225354026041706724885718, −2.72563216463154576792965327233, −1.89799947942920710250432525691,
0.66608343309011647481943531964, 1.74301392214192615158583794373, 3.28591396655418470353751689134, 3.88406399621896925000348616399, 4.62975403998748206643377002629, 6.00604711700862598747778407946, 6.74479230578578141503988159195, 7.88634461345910910327856633476, 8.146509811382903515537619200586, 8.964314357602011558765948777096