L(s) = 1 | − 2-s + (1.68 − 0.396i)3-s + 4-s + (−2.18 − 0.469i)5-s + (−1.68 + 0.396i)6-s − 8-s + (2.68 − 1.33i)9-s + (2.18 + 0.469i)10-s + 0.939i·11-s + (1.68 − 0.396i)12-s − 2·13-s + (−3.87 + 0.0737i)15-s + 16-s + 6.63i·17-s + (−2.68 + 1.33i)18-s + 3.46i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.973 − 0.228i)3-s + 0.5·4-s + (−0.977 − 0.210i)5-s + (−0.688 + 0.161i)6-s − 0.353·8-s + (0.895 − 0.445i)9-s + (0.691 + 0.148i)10-s + 0.283i·11-s + (0.486 − 0.114i)12-s − 0.554·13-s + (−0.999 + 0.0190i)15-s + 0.250·16-s + 1.60i·17-s + (−0.633 + 0.314i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.289757993\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289757993\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.68 + 0.396i)T \) |
| 5 | \( 1 + (2.18 + 0.469i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 0.939iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 3.31iT - 29T^{2} \) |
| 31 | \( 1 - 7.57iT - 31T^{2} \) |
| 37 | \( 1 + 8.21iT - 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 1.08iT - 43T^{2} \) |
| 47 | \( 1 + 8.51iT - 47T^{2} \) |
| 53 | \( 1 - 4.37T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 2.37iT - 67T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 - 1.37T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−9.413359429521390831447501931510, −8.552809632454748677198472087766, −8.244695060134303491501650131977, −7.31665743582107399790713581373, −6.86312767863944714342216889134, −5.52923623971073379891182565642, −4.18297389085895659119969362874, −3.54860932470978812693029711503, −2.35510228566111763114089670978, −1.21512007577811582132652547180,
0.64818794872419168179988565399, 2.44751314752646849657416039542, 3.05536164915501303380330277931, 4.19672429070276018582710666226, 5.02990051690629300453464516254, 6.55435547402494762405103489802, 7.37762608391294887578434165766, 7.80120820153681778522814282596, 8.616859828738974552202868244450, 9.406212887781426186597201423619