| L(s) = 1 | − i·3-s + 3.91i·5-s + i·7-s − 9-s + 4.44i·11-s + 4.13·13-s + 3.91·15-s + (−2.43 − 3.32i)17-s − 3.38·19-s + 21-s + 2.97i·23-s − 10.3·25-s + i·27-s + 1.24i·29-s − 7.09i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.74i·5-s + 0.377i·7-s − 0.333·9-s + 1.33i·11-s + 1.14·13-s + 1.01·15-s + (−0.590 − 0.807i)17-s − 0.775·19-s + 0.218·21-s + 0.619i·23-s − 2.06·25-s + 0.192i·27-s + 0.231i·29-s − 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203097263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203097263\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 17 | \( 1 + (2.43 + 3.32i)T \) |
| good | 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 11 | \( 1 - 4.44iT - 11T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 - 2.97iT - 23T^{2} \) |
| 29 | \( 1 - 1.24iT - 29T^{2} \) |
| 31 | \( 1 + 7.09iT - 31T^{2} \) |
| 37 | \( 1 + 0.739iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 - 12.8iT - 61T^{2} \) |
| 67 | \( 1 - 4.86T + 67T^{2} \) |
| 71 | \( 1 + 6.42iT - 71T^{2} \) |
| 73 | \( 1 + 0.225iT - 73T^{2} \) |
| 79 | \( 1 + 17.2iT - 79T^{2} \) |
| 83 | \( 1 + 3.39T + 83T^{2} \) |
| 89 | \( 1 - 8.44T + 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−9.874543293662321413897712816032, −9.094998077809834880506920007425, −7.927774732614447007955121232941, −7.36158576153406726027713545905, −6.46545329819561214237172300319, −6.19179508635730699744545972496, −4.77565207269280028299905967447, −3.60434939660468805292470081391, −2.64695181995839170389577764341, −1.83688732705071338618835854935,
0.47544567616452439385558932220, 1.67583916061758876541529305365, 3.45491800137513742552013773458, 4.15880658236385028364815011855, 4.98701246961903211900210323218, 5.82611742357423727062615844171, 6.57525061961559459414366824907, 8.240108876919324380448460329231, 8.476560798451251014694707946431, 8.986545930821161979946953999801
