| L(s) = 1 | − 2.46·2-s − 3-s + 4.07·4-s − 0.653·5-s + 2.46·6-s + 7-s − 5.11·8-s + 9-s + 1.61·10-s − 2.42·11-s − 4.07·12-s + 4.26·13-s − 2.46·14-s + 0.653·15-s + 4.46·16-s + 2.38·17-s − 2.46·18-s − 5.35·19-s − 2.66·20-s − 21-s + 5.97·22-s − 23-s + 5.11·24-s − 4.57·25-s − 10.5·26-s − 27-s + 4.07·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.03·4-s − 0.292·5-s + 1.00·6-s + 0.377·7-s − 1.80·8-s + 0.333·9-s + 0.509·10-s − 0.730·11-s − 1.17·12-s + 1.18·13-s − 0.658·14-s + 0.168·15-s + 1.11·16-s + 0.579·17-s − 0.581·18-s − 1.22·19-s − 0.595·20-s − 0.218·21-s + 1.27·22-s − 0.208·23-s + 1.04·24-s − 0.914·25-s − 2.06·26-s − 0.192·27-s + 0.770·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4804177490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4804177490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 + 0.653T + 5T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.388T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 - 5.73T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 2.00T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 1.73T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 - 6.07T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 - 0.0112T + 97T^{2} \) |
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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
−10.87457300809835599177114199429, −10.14330461838397057661387284795, −9.194822698373291854396878260032, −8.172047621097236342495312124326, −7.78139986995150934986569728195, −6.58120269401366277617222339993, −5.73534552075346854950231378122, −4.13830050850588763243702257286, −2.32691015753255618927708196658, −0.853408866777955455298082469435,
0.853408866777955455298082469435, 2.32691015753255618927708196658, 4.13830050850588763243702257286, 5.73534552075346854950231378122, 6.58120269401366277617222339993, 7.78139986995150934986569728195, 8.172047621097236342495312124326, 9.194822698373291854396878260032, 10.14330461838397057661387284795, 10.87457300809835599177114199429
