L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (0.433 − 0.900i)8-s + (0.365 + 0.930i)11-s + (0.930 − 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (−0.5 + 0.866i)19-s + (0.294 − 0.955i)22-s + (−0.680 − 0.733i)23-s + (−0.955 − 0.294i)26-s + (−0.955 + 0.294i)29-s − 31-s + (0.974 + 0.222i)32-s + (0.365 − 0.930i)34-s + ⋯ |
L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (0.433 − 0.900i)8-s + (0.365 + 0.930i)11-s + (0.930 − 0.365i)13-s + (−0.900 + 0.433i)16-s + (0.294 + 0.955i)17-s + (−0.5 + 0.866i)19-s + (0.294 − 0.955i)22-s + (−0.680 − 0.733i)23-s + (−0.955 − 0.294i)26-s + (−0.955 + 0.294i)29-s − 31-s + (0.974 + 0.222i)32-s + (0.365 − 0.930i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.08571560552 + 0.2575241797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08571560552 + 0.2575241797i\) |
\(L(1)\) |
\(\approx\) |
\(0.6306475990 - 0.03899356191i\) |
\(L(1)\) |
\(\approx\) |
\(0.6306475990 - 0.03899356191i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 11 | \( 1 + (0.365 + 0.930i)T \) |
| 13 | \( 1 + (0.930 - 0.365i)T \) |
| 17 | \( 1 + (0.294 + 0.955i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.680 - 0.733i)T \) |
| 29 | \( 1 + (-0.955 + 0.294i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.680 + 0.733i)T \) |
| 41 | \( 1 + (-0.826 + 0.563i)T \) |
| 43 | \( 1 + (-0.563 + 0.826i)T \) |
| 47 | \( 1 + (0.781 + 0.623i)T \) |
| 53 | \( 1 + (-0.680 - 0.733i)T \) |
| 59 | \( 1 + (-0.900 + 0.433i)T \) |
| 61 | \( 1 + (0.222 - 0.974i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (-0.930 - 0.365i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (-0.930 - 0.365i)T \) |
| 89 | \( 1 + (-0.988 + 0.149i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.16610018778367248435902205425, −18.69432815716931736511142899160, −18.06474860292303764881502589358, −17.17045758418818612415756791884, −16.64600859783757964849024126405, −15.84028334214951323514105442144, −15.44625611370439393348808156761, −14.33987707483343148365547526363, −13.85621614892087567079956726722, −13.15207365618013157857895548855, −11.792246948058251840339903979909, −11.28469383039160013049547481767, −10.59410054773428801228685541005, −9.66104533780465759037571032934, −8.87236065443483563259621700853, −8.549857565491150420825207949128, −7.3744473433363188246478801382, −6.93863689195338252409967085549, −5.83807113418116605894656633346, −5.50993832548790612534263098640, −4.26237458189329105798730117867, −3.35032044950368383431172141829, −2.14321189633244990036359112108, −1.2559343677097385260357163065, −0.11814361833786726460284284775,
1.55941743415618869618051560633, 1.781551108769977449564008580930, 3.16433871200126563956776390692, 3.80551129671866057922675824438, 4.61348755318203268254805844374, 5.942362760463403376103660871491, 6.61871631027718135773659788538, 7.626628149177364266759593717727, 8.24008009815599548909878758046, 8.94088416171733968585649003874, 9.83363521393059300200779994940, 10.4178255969691111914429880507, 11.07040338106830514021700543241, 11.974691688919935089365169909021, 12.642744180655652481750006278213, 13.11736716280212095006547447264, 14.26748389628688739839813561774, 15.02949537766636567531384320045, 15.830814446956646443716993338563, 16.74087350097521642606173612373, 17.10055408614232989343309550782, 18.10480980646094693616873569533, 18.48450904456935824786180954434, 19.26026289888280961391177856721, 20.140187804842711758054482669984