L(s) = 1 | − 2-s + (−0.597 − 0.802i)3-s + 4-s + (−0.230 + 0.973i)5-s + (0.597 + 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.597 − 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.918 − 0.396i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.597 − 0.802i)3-s + 4-s + (−0.230 + 0.973i)5-s + (0.597 + 0.802i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (0.230 − 0.973i)10-s + (−0.230 − 0.973i)11-s + (−0.597 − 0.802i)12-s + (−0.973 − 0.230i)13-s + (0.286 + 0.957i)14-s + (0.918 − 0.396i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02870808827 - 0.1977420880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02870808827 - 0.1977420880i\) |
\(L(1)\) |
\(\approx\) |
\(0.4335384601 - 0.1333588108i\) |
\(L(1)\) |
\(\approx\) |
\(0.4335384601 - 0.1333588108i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.597 - 0.802i)T \) |
| 5 | \( 1 + (-0.230 + 0.973i)T \) |
| 7 | \( 1 + (-0.286 - 0.957i)T \) |
| 11 | \( 1 + (-0.230 - 0.973i)T \) |
| 13 | \( 1 + (-0.973 - 0.230i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.957 - 0.286i)T \) |
| 31 | \( 1 + (-0.448 + 0.893i)T \) |
| 41 | \( 1 + (0.984 + 0.173i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.802 - 0.597i)T \) |
| 53 | \( 1 + (0.116 + 0.993i)T \) |
| 59 | \( 1 + (-0.396 - 0.918i)T \) |
| 61 | \( 1 + (0.549 - 0.835i)T \) |
| 67 | \( 1 + (-0.802 - 0.597i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.286 - 0.957i)T \) |
| 79 | \( 1 + (-0.396 - 0.918i)T \) |
| 83 | \( 1 + (0.0581 - 0.998i)T \) |
| 89 | \( 1 + (0.727 + 0.686i)T \) |
| 97 | \( 1 + (0.448 + 0.893i)T \) |
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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
−18.736921917092020894553361984219, −17.91039897355865599900530693011, −17.480348831996898879650500296964, −16.839847027945077713544178705994, −16.12239831548344906493919500115, −15.746982356060846963663838810866, −15.04700974190546212148031190064, −14.55542388408107932613803701991, −12.887903440463939700454017991130, −12.47675583793110069439138945635, −11.85582718915853528680839176383, −11.33730053046318138059358571105, −10.24925002724146958911328951961, −9.88437179966788677233534202070, −9.03123823907861935078068340297, −8.82826239568853528919833802384, −7.80768505006385168264089213121, −6.99274393947887319078099903201, −6.154732173347318774971554506211, −5.41331598974530314394244481846, −4.76640067431485271706785802362, −3.97649855007740842910646808469, −2.77575094538585473112833069694, −2.11583338874876251195556263773, −0.97661171152982779908272228476,
0.12076145175383220219827243477, 0.89850850477660203330463450260, 1.88326958085706536609253102128, 2.99159262841385548695504433766, 3.19656011807831603472224197204, 4.71682290763823543501138943330, 5.70295249324113847662776509554, 6.415347012005328316056614626758, 7.03065291486204421148313626731, 7.634191089815373092201789228982, 7.90626592508807225410668893173, 9.1503367860924845537020237217, 9.95672603357967301763246957830, 10.55793296276103373894539752673, 11.17214569467154876782097057523, 11.647064280710270228610739713788, 12.372519054349044047956042089160, 13.37683374536260880958998135, 13.98886310781127950367967246315, 14.62633907876699559241112813753, 15.847607020530701047266101464846, 16.10839825440179159820476674827, 16.89112096475034640990494538698, 17.75440681345159121431923820117, 17.89290942453350965955711159901