| L(s) = 1 | − i·2-s + (−1.67 + 0.449i)3-s − 4-s − 5-s + (0.449 + 1.67i)6-s + i·8-s + (2.59 − 1.50i)9-s + i·10-s − 2.37i·11-s + (1.67 − 0.449i)12-s + 3.07i·13-s + (1.67 − 0.449i)15-s + 16-s − 4.70·17-s + (−1.50 − 2.59i)18-s − 0.527i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.965 + 0.259i)3-s − 0.5·4-s − 0.447·5-s + (0.183 + 0.682i)6-s + 0.353i·8-s + (0.865 − 0.501i)9-s + 0.316i·10-s − 0.717i·11-s + (0.482 − 0.129i)12-s + 0.852i·13-s + (0.431 − 0.116i)15-s + 0.250·16-s − 1.14·17-s + (−0.354 − 0.611i)18-s − 0.120i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8586178584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8586178584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.67 - 0.449i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2.37iT - 11T^{2} \) |
| 13 | \( 1 - 3.07iT - 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 + 0.527iT - 19T^{2} \) |
| 23 | \( 1 - 4.62iT - 23T^{2} \) |
| 29 | \( 1 + 0.405iT - 29T^{2} \) |
| 31 | \( 1 - 1.02iT - 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 8.61T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 + 7.51iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 - 8.65iT - 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 19.6iT - 97T^{2} \) |
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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
−9.362984477560980926115754140334, −8.970128109752560254074816291514, −7.78414188870805299985245937160, −6.85473990106261387029850055874, −6.02142550651840226378019305950, −5.07298982563349288489531250951, −4.23823756662110862422931188585, −3.52108279710332552125580208115, −2.06437520836450601608665793124, −0.64529370132985968692820939006,
0.73638105684128272344337836360, 2.41809222094656450061682683719, 4.08583471081687856154152760023, 4.66631239338675700171394948870, 5.63944959683761848420486730887, 6.34907203730127776617292214945, 7.22723078390352941809626198576, 7.69146195304875440803226730422, 8.705459239941012199860109074837, 9.545853633823975670758835996520
