L(s) = 1 | − 3.17·3-s − i·5-s + 7.10·9-s − 1.11i·11-s − 3.17i·13-s + 3.17i·15-s − 6.37i·17-s + 8.30·19-s + 6.55i·23-s − 25-s − 13.0·27-s + 7.40·29-s + 1.53·31-s + 3.55i·33-s − 0.0147·37-s + ⋯ |
L(s) = 1 | − 1.83·3-s − 0.447i·5-s + 2.36·9-s − 0.336i·11-s − 0.881i·13-s + 0.820i·15-s − 1.54i·17-s + 1.90·19-s + 1.36i·23-s − 0.200·25-s − 2.51·27-s + 1.37·29-s + 0.276·31-s + 0.617i·33-s − 0.00243·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.039575138\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.039575138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 3.17T + 3T^{2} \) |
| 11 | \( 1 + 1.11iT - 11T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 + 6.37iT - 17T^{2} \) |
| 19 | \( 1 - 8.30T + 19T^{2} \) |
| 23 | \( 1 - 6.55iT - 23T^{2} \) |
| 29 | \( 1 - 7.40T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 + 0.0147T + 37T^{2} \) |
| 41 | \( 1 + 1.89iT - 41T^{2} \) |
| 43 | \( 1 + 6.07iT - 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 + 5.50T + 59T^{2} \) |
| 61 | \( 1 + 8.26iT - 61T^{2} \) |
| 67 | \( 1 - 4.38iT - 67T^{2} \) |
| 71 | \( 1 - 14.8iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 - 6.29iT - 79T^{2} \) |
| 83 | \( 1 + 7.10T + 83T^{2} \) |
| 89 | \( 1 - 11.0iT - 89T^{2} \) |
| 97 | \( 1 - 15.3iT - 97T^{2} \) |
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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
−8.182887875026858683413296267702, −7.30394490234587361202336455336, −6.92479529850654334418751967237, −5.80156319327346906344092701638, −5.34355280177353730257993879303, −5.00118647003951942271085869489, −3.91020688705622777664780800774, −2.84725665155424365872490554818, −1.16827711710905606482989486634, −0.63164158408516719211548398004,
0.857415588537072648385145607103, 1.85211254344958481526019937442, 3.25583396691470962854394042305, 4.45289793365027575602205443735, 4.73322144995122781236219883840, 5.91208210319444676148963883313, 6.19617917369598110198277844950, 6.96015005849013754408581295759, 7.55097638039987759183973327440, 8.584251747267803807181197970161