| L(s) = 1 | − i·2-s − 1.61i·3-s − 4-s − 2.23·5-s − 1.61·6-s − 1.85i·7-s + i·8-s + 0.381·9-s + 2.23i·10-s − 5.61·11-s + 1.61i·12-s − 2.61i·13-s − 1.85·14-s + 3.61i·15-s + 16-s + 0.854i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.934i·3-s − 0.5·4-s − 0.999·5-s − 0.660·6-s − 0.700i·7-s + 0.353i·8-s + 0.127·9-s + 0.707i·10-s − 1.69·11-s + 0.467i·12-s − 0.726i·13-s − 0.495·14-s + 0.934i·15-s + 0.250·16-s + 0.207i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.759768i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759768i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + 2.23T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.61iT - 3T^{2} \) |
| 7 | \( 1 + 1.85iT - 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 + 2.61iT - 13T^{2} \) |
| 17 | \( 1 - 0.854iT - 17T^{2} \) |
| 19 | \( 1 - 0.145T + 19T^{2} \) |
| 29 | \( 1 - 9.70T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9.70iT - 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 1.70iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 5.23iT - 67T^{2} \) |
| 71 | \( 1 - 0.381T + 71T^{2} \) |
| 73 | \( 1 - 16.4iT - 73T^{2} \) |
| 79 | \( 1 - 7.70T + 79T^{2} \) |
| 83 | \( 1 + 7.70iT - 83T^{2} \) |
| 89 | \( 1 + 3.70T + 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−11.91871493687861706509599897548, −10.65893706578457776771152062713, −10.29130766873002452587857503702, −8.486014673715489236286559024824, −7.76983747648852652437570480047, −6.99715725445727066908852194873, −5.27036176480170043043237402166, −3.97501786234077577623805799731, −2.58276374650143322805870526580, −0.63326505294999348003538572031,
3.09085984968487180357932001720, 4.52986418414509946267787643482, 5.16174315681978554591801592176, 6.69561740979144057830978451218, 7.86321153435029193482324787266, 8.617506485191786541322690274082, 9.747225826089862165706865860619, 10.61080730296115361101711821500, 11.70852398378159037070945200787, 12.68034143927252445119929055234
