L(s) = 1 | + (0.876 − 0.481i)2-s + (0.637 + 0.770i)3-s + (0.535 − 0.844i)4-s + (0.929 + 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (0.992 − 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (0.929 + 0.368i)19-s + (−0.0627 − 0.998i)21-s + ⋯ |
L(s) = 1 | + (0.876 − 0.481i)2-s + (0.637 + 0.770i)3-s + (0.535 − 0.844i)4-s + (0.929 + 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (0.992 − 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (0.929 + 0.368i)19-s + (−0.0627 − 0.998i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.349420942 - 2.332749222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.349420942 - 2.332749222i\) |
\(L(1)\) |
\(\approx\) |
\(1.709411747 - 0.4036549796i\) |
\(L(1)\) |
\(\approx\) |
\(1.709411747 - 0.4036549796i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.876 - 0.481i)T \) |
| 3 | \( 1 + (0.637 + 0.770i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.187 + 0.982i)T \) |
| 17 | \( 1 + (-0.637 + 0.770i)T \) |
| 19 | \( 1 + (0.929 + 0.368i)T \) |
| 23 | \( 1 + (0.425 - 0.904i)T \) |
| 29 | \( 1 + (-0.968 - 0.248i)T \) |
| 31 | \( 1 + (0.0627 - 0.998i)T \) |
| 37 | \( 1 + (0.187 - 0.982i)T \) |
| 41 | \( 1 + (0.187 - 0.982i)T \) |
| 43 | \( 1 + (-0.809 + 0.587i)T \) |
| 47 | \( 1 + (0.929 - 0.368i)T \) |
| 53 | \( 1 + (0.929 - 0.368i)T \) |
| 59 | \( 1 + (0.728 - 0.684i)T \) |
| 61 | \( 1 + (-0.728 - 0.684i)T \) |
| 67 | \( 1 + (0.637 - 0.770i)T \) |
| 71 | \( 1 + (-0.637 - 0.770i)T \) |
| 73 | \( 1 + (0.876 - 0.481i)T \) |
| 79 | \( 1 + (-0.0627 - 0.998i)T \) |
| 83 | \( 1 + (0.0627 - 0.998i)T \) |
| 89 | \( 1 + (-0.992 - 0.125i)T \) |
| 97 | \( 1 + (-0.535 + 0.844i)T \) |
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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
−20.88366952228181781217336365773, −20.02974721886749717680476091913, −19.73647607160965789529820148500, −18.51209010990278754572447071736, −17.9665453944536742674427326474, −17.08318048001751720103243195797, −16.06148046946962342018826328846, −15.364281717984850462541158722073, −14.924190004272591496381696726044, −13.78030827030325034058206759050, −13.37493108322407342790698658289, −12.68899437971590717878676332196, −11.97880253463024435686597227863, −11.25493558708054479123471687324, −9.83155697970195067507186798554, −8.99684048458848258389451185710, −8.23168545658956450311572143634, −7.21419521184570746321301919121, −6.88936400078407969277747019958, −5.76149033516019273277625531876, −5.21123625537486186919405699936, −3.8381785114100064778038000281, −2.93798011253429323177006356771, −2.61632145778954559404688575188, −1.19305342213833148884811862491,
0.315520559850187871130238740956, 1.821482273751601362250395011023, 2.593545680627585769972607551213, 3.702891958296817297632947645115, 4.005053401017090098140731870861, 4.95159326753816712040664281795, 5.95466812620646774337322746881, 6.84041720265799833716178148267, 7.717348142540692049076868697453, 9.076398101192217973583565504113, 9.56738866370962430496880880021, 10.43263581064668504070750875638, 11.00674410763010008314751062743, 11.95583574114860092201540652576, 12.968677008816240294600411723985, 13.51571347197150324457277565240, 14.26090405808032427172265538063, 14.90534055537662693731284221414, 15.70214454785678589655943562353, 16.4227829405143791877155601292, 16.98305575801577295427408834863, 18.57669822926750369627204239875, 19.18668143043287184620747191391, 19.88913982008140579412288177798, 20.44822188136945956913781223487