| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 3.04i·7-s − i·8-s + 9-s − 10-s − 1.13i·11-s + 12-s + 3.04·14-s − i·15-s + 16-s − 2.69·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.15i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 0.342i·11-s + 0.288·12-s + 0.814·14-s − 0.258i·15-s + 0.250·16-s − 0.652·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013384847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013384847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 + 1.13iT - 11T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 - 1.55iT - 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 + 6.15T + 29T^{2} \) |
| 31 | \( 1 + 0.246iT - 31T^{2} \) |
| 37 | \( 1 - 0.554iT - 37T^{2} \) |
| 41 | \( 1 - 8.26iT - 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 6.70iT - 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 + 1.36iT - 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 2.47iT - 67T^{2} \) |
| 71 | \( 1 + 8.57iT - 71T^{2} \) |
| 73 | \( 1 - 14.8iT - 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 + 5.45iT - 83T^{2} \) |
| 89 | \( 1 + 16.8iT - 89T^{2} \) |
| 97 | \( 1 - 1.93iT - 97T^{2} \) |
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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
−8.114263465453249525867603595683, −7.53752684091129744947990244832, −6.93298616429173893920606194878, −6.24998407789623265923805909975, −5.70951838807116753231821811839, −4.70267618663335665094006230453, −4.06911403856038903403681610501, −3.35504845592714652108366990047, −1.94618358620296586297386764952, −0.68389633364134249059480563018,
0.45154405976837365496967814657, 1.89392803709015107334140406139, 2.33707369050268072767400543576, 3.63252999630653618805458313009, 4.33257777362928464317868344958, 5.27366552222166158009247037677, 5.60784030012467952149453910987, 6.52297014861590520803285429172, 7.38411864789853440506407852471, 8.314162379047258734504016438795
