L(s) = 1 | + (0.281 − 0.959i)3-s + (−0.142 + 0.989i)7-s + (−0.841 − 0.540i)9-s + (0.909 − 0.415i)11-s + (−0.989 + 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (0.909 + 0.415i)21-s + (−0.755 + 0.654i)27-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (−0.142 − 0.989i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (−0.841 + 0.540i)41-s + ⋯ |
L(s) = 1 | + (0.281 − 0.959i)3-s + (−0.142 + 0.989i)7-s + (−0.841 − 0.540i)9-s + (0.909 − 0.415i)11-s + (−0.989 + 0.142i)13-s + (−0.654 + 0.755i)17-s + (0.755 − 0.654i)19-s + (0.909 + 0.415i)21-s + (−0.755 + 0.654i)27-s + (−0.755 − 0.654i)29-s + (0.959 − 0.281i)31-s + (−0.142 − 0.989i)33-s + (0.540 − 0.841i)37-s + (−0.142 + 0.989i)39-s + (−0.841 + 0.540i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.060108183 - 1.028933521i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060108183 - 1.028933521i\) |
\(L(1)\) |
\(\approx\) |
\(1.039306921 - 0.3495547619i\) |
\(L(1)\) |
\(\approx\) |
\(1.039306921 - 0.3495547619i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.281 - 0.959i)T \) |
| 7 | \( 1 + (-0.142 + 0.989i)T \) |
| 11 | \( 1 + (0.909 - 0.415i)T \) |
| 13 | \( 1 + (-0.989 + 0.142i)T \) |
| 17 | \( 1 + (-0.654 + 0.755i)T \) |
| 19 | \( 1 + (0.755 - 0.654i)T \) |
| 29 | \( 1 + (-0.755 - 0.654i)T \) |
| 31 | \( 1 + (0.959 - 0.281i)T \) |
| 37 | \( 1 + (0.540 - 0.841i)T \) |
| 41 | \( 1 + (-0.841 + 0.540i)T \) |
| 43 | \( 1 + (0.281 - 0.959i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.989 + 0.142i)T \) |
| 59 | \( 1 + (0.989 - 0.142i)T \) |
| 61 | \( 1 + (0.281 + 0.959i)T \) |
| 67 | \( 1 + (-0.909 - 0.415i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.654 - 0.755i)T \) |
| 79 | \( 1 + (-0.142 - 0.989i)T \) |
| 83 | \( 1 + (0.540 - 0.841i)T \) |
| 89 | \( 1 + (-0.959 - 0.281i)T \) |
| 97 | \( 1 + (0.841 - 0.540i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.23434114816654622881358254185, −19.90826370520777633008474036651, −19.09334327703563357159531817520, −17.954878331934211936852510712325, −17.13557309604823884819118724037, −16.72488632662965159361196734827, −15.93955612872897802217892276379, −15.1444175260757924138978637961, −14.36850965157621779802207531921, −13.94404568372536320406704068005, −13.02683165047019856764703397931, −11.928363383587875438301601231282, −11.35873985831430645184996959968, −10.35365891573178907273611222418, −9.82748002317426407686878084672, −9.25374150187365121498031240802, −8.27794627518638221571929848719, −7.34764329761228872630373243362, −6.73620025724582435991133491938, −5.50726371988532204385453878157, −4.69500458208780233061295902608, −4.06524815583852327344809506730, −3.26042989728627648152040086717, −2.32614052360574758264567253406, −1.0254636310617265097493722999,
0.56696979361272932322536924342, 1.83897872680240407901535011802, 2.45542824875883827684328399720, 3.34435782731343914944369584587, 4.432803345043367634425889941, 5.60956928577009929681770749393, 6.17538169357962723029441660180, 7.016547375256140900109124932177, 7.733781862003879065673323239943, 8.81370923659284247357188098187, 9.046570895860533310374079463728, 10.06911784194712529079521705948, 11.399543039531575737942104671451, 11.799235153493148878804493117, 12.47983078301594395616549969008, 13.312241483442772830786725589360, 13.930556581154834205891543451894, 14.916888919774450505718458996727, 15.212685918245873273635612339860, 16.41148513418964701982057888781, 17.23610629418057813546231133579, 17.7741742680904764581753381263, 18.61542135402433111861145531277, 19.327958244633279462932857135201, 19.64349947195756265806285231303