L(s) = 1 | − 2.40·2-s + (−1.54 + 0.777i)3-s + 3.79·4-s + (3.72 − 1.87i)6-s + (2.44 + i)7-s − 4.31·8-s + (1.79 − 2.40i)9-s + 4.31i·11-s + (−5.86 + 2.94i)12-s + 2.44·13-s + (−5.89 − 2.40i)14-s + 2.79·16-s − 5.89i·17-s + (−4.31 + 5.79i)18-s − 6.83i·19-s + ⋯ |
L(s) = 1 | − 1.70·2-s + (−0.893 + 0.448i)3-s + 1.89·4-s + (1.52 − 0.763i)6-s + (0.925 + 0.377i)7-s − 1.52·8-s + (0.597 − 0.802i)9-s + 1.29i·11-s + (−1.69 + 0.850i)12-s + 0.679·13-s + (−1.57 − 0.643i)14-s + 0.697·16-s − 1.42i·17-s + (−1.01 + 1.36i)18-s − 1.56i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.521783 + 0.144832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.521783 + 0.144832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.54 - 0.777i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.44 - i)T \) |
good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 11 | \( 1 - 4.31iT - 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 5.89iT - 17T^{2} \) |
| 19 | \( 1 + 6.83iT - 19T^{2} \) |
| 23 | \( 1 - 0.502T + 23T^{2} \) |
| 29 | \( 1 + 0.502iT - 29T^{2} \) |
| 31 | \( 1 - 4.38iT - 31T^{2} \) |
| 37 | \( 1 - 0.582iT - 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 - 2.58iT - 43T^{2} \) |
| 47 | \( 1 + 5.89iT - 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 4.38iT - 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.31iT - 71T^{2} \) |
| 73 | \( 1 + 6.32T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 7.12iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 0.511T + 97T^{2} \) |
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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
−10.81699742468492039447069691040, −9.967866577645560951829684022586, −9.242782245918759578878557473477, −8.548250999481044845483736144532, −7.25172922166600199610099357728, −6.87262794918433440084490342703, −5.38151269163192447729406688566, −4.49145778969072228539772277083, −2.40135751821303900844923399628, −0.957227517005743187538836950826,
0.912775764673700050471878979403, 1.86684791248305600908314874923, 3.96365160303542647759359526130, 5.70150252124976690940402949662, 6.30738810050763710557421920229, 7.54618924616950836479164917187, 8.127981042522129306567304014969, 8.765306977785515931908839063142, 10.17946141774601265835818436423, 10.69287515449828002174774509868