L(s) = 1 | + (−0.725 + 1.21i)2-s + (−0.946 − 1.76i)4-s + 3.06i·5-s − 7-s + (2.82 + 0.130i)8-s + (−3.72 − 2.22i)10-s − 5.80i·11-s + 3.52i·13-s + (0.725 − 1.21i)14-s + (−2.20 + 3.33i)16-s + 6.79·17-s − 5.28i·19-s + (5.40 − 2.90i)20-s + (7.04 + 4.21i)22-s + 5.65·23-s + ⋯ |
L(s) = 1 | + (−0.513 + 0.858i)2-s + (−0.473 − 0.880i)4-s + 1.37i·5-s − 0.377·7-s + (0.998 + 0.0460i)8-s + (−1.17 − 0.704i)10-s − 1.75i·11-s + 0.977i·13-s + (0.193 − 0.324i)14-s + (−0.552 + 0.833i)16-s + 1.64·17-s − 1.21i·19-s + (1.20 − 0.649i)20-s + (1.50 + 0.898i)22-s + 1.17·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0460 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0460 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.221885162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221885162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.725 - 1.21i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 3.06iT - 5T^{2} \) |
| 11 | \( 1 + 5.80iT - 11T^{2} \) |
| 13 | \( 1 - 3.52iT - 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 + 5.28iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.21iT - 29T^{2} \) |
| 31 | \( 1 - 0.107T + 31T^{2} \) |
| 37 | \( 1 - 4.90iT - 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 1.85iT - 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + 7.85iT - 59T^{2} \) |
| 61 | \( 1 + 12.0iT - 61T^{2} \) |
| 67 | \( 1 - 6.66iT - 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 + 6.52T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−9.490345824338296741959192264001, −8.902985626571017270084047646998, −7.927459240607901749117863542828, −7.19158176085519615876423734171, −6.47316865788794658396220357334, −5.95617080520689987938095016450, −4.91659957233708011093522650581, −3.53306916629409168023559421879, −2.78065375442855659265928124100, −0.954412163671556532903735941540,
0.821706155197072658631174586551, 1.78574651509908858551018899872, 3.09937196271238162781989032964, 4.11762360660082688318645492629, 4.94760840066310573486384669081, 5.72458528181973677387572018934, 7.34149240857500282292008950729, 7.77951755864335816435317223085, 8.655026905193506723522015665821, 9.457465752428214036867854227029