| L(s) = 1 | − 1.19·2-s − 2.00i·3-s − 0.565·4-s + 1.69i·5-s + 2.40i·6-s − 3.98i·7-s + 3.07·8-s − 1.03·9-s − 2.03i·10-s − 1.80i·11-s + 1.13i·12-s + 4.43·13-s + 4.76i·14-s + 3.41·15-s − 2.54·16-s + (−1.94 + 3.63i)17-s + ⋯ |
| L(s) = 1 | − 0.846·2-s − 1.15i·3-s − 0.282·4-s + 0.759i·5-s + 0.981i·6-s − 1.50i·7-s + 1.08·8-s − 0.344·9-s − 0.643i·10-s − 0.543i·11-s + 0.327i·12-s + 1.23·13-s + 1.27i·14-s + 0.880·15-s − 0.637·16-s + (−0.472 + 0.881i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.452864 - 0.756807i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452864 - 0.756807i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.94 - 3.63i)T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 3 | \( 1 + 2.00iT - 3T^{2} \) |
| 5 | \( 1 - 1.69iT - 5T^{2} \) |
| 7 | \( 1 + 3.98iT - 7T^{2} \) |
| 11 | \( 1 + 1.80iT - 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 19 | \( 1 - 4.57T + 19T^{2} \) |
| 23 | \( 1 + 1.17iT - 23T^{2} \) |
| 29 | \( 1 - 6.34iT - 29T^{2} \) |
| 31 | \( 1 + 8.90iT - 31T^{2} \) |
| 37 | \( 1 - 0.226iT - 37T^{2} \) |
| 41 | \( 1 + 8.29iT - 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 53 | \( 1 + 9.34T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 5.79iT - 71T^{2} \) |
| 73 | \( 1 - 6.00iT - 73T^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 0.226T + 89T^{2} \) |
| 97 | \( 1 - 0.662iT - 97T^{2} \) |
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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
−10.09516278455193138957070437374, −9.008169568700383622544098308648, −8.110118913044622433704554724236, −7.47057764818651213463614235258, −6.86718633534135534098399517408, −5.98934300442856248882166753965, −4.34000504740916002716691085741, −3.40605911286701885387315537927, −1.61853234945605719148994383960, −0.70533835891322160736385093538,
1.40280726814018216193936797058, 3.09935760542151486839694304689, 4.49214660677052454535740083077, 4.96405093810607320311725576078, 5.94134487567063147016960642908, 7.42424347547581019145641886213, 8.530395763325758190376125382994, 9.019692077979226750784023862412, 9.437690188944697489590974983245, 10.21538067974437516725017424038
