L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7962311769\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7962311769\) |
\(L(1)\) |
\(\approx\) |
\(0.5602448972\) |
\(L(1)\) |
\(\approx\) |
\(0.5602448972\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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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
−25.31242222675621664431551739755, −24.36610850506113872494423921514, −23.72258772846889737475407982539, −22.83723049365914314285573862869, −21.592323187791709648421065967413, −20.742393195445823091931121159737, −19.67382395362042506777090354142, −18.85039000295176749194293212797, −17.91300960854373126727963838780, −17.24394728289450202837672884657, −16.332277647475108549071836492664, −15.50514563989933891874055081686, −14.65436184856480236258973630043, −12.82744108125370481899881127449, −11.745963593102807174415108490889, −11.18778413704913020983330402150, −10.620249579466489680492312424333, −9.00335779439256028902621030802, −8.27094561702072828724535719141, −7.05196956742698477295442976043, −6.36740108023306933200730068547, −4.85415067204300359203549781994, −3.73900693182454968811093040255, −1.74695240500948440151273778144, −0.69556002559801935321157760296,
0.69556002559801935321157760296, 1.74695240500948440151273778144, 3.73900693182454968811093040255, 4.85415067204300359203549781994, 6.36740108023306933200730068547, 7.05196956742698477295442976043, 8.27094561702072828724535719141, 9.00335779439256028902621030802, 10.620249579466489680492312424333, 11.18778413704913020983330402150, 11.745963593102807174415108490889, 12.82744108125370481899881127449, 14.65436184856480236258973630043, 15.50514563989933891874055081686, 16.332277647475108549071836492664, 17.24394728289450202837672884657, 17.91300960854373126727963838780, 18.85039000295176749194293212797, 19.67382395362042506777090354142, 20.742393195445823091931121159737, 21.592323187791709648421065967413, 22.83723049365914314285573862869, 23.72258772846889737475407982539, 24.36610850506113872494423921514, 25.31242222675621664431551739755