| L(s) = 1 | + 4.21·2-s + 3·3-s + 9.79·4-s − 19.6·5-s + 12.6·6-s + 7·7-s + 7.58·8-s + 9·9-s − 82.7·10-s − 40.2·11-s + 29.3·12-s + 19.9·13-s + 29.5·14-s − 58.8·15-s − 46.3·16-s − 96.9·17-s + 37.9·18-s − 124.·19-s − 192.·20-s + 21·21-s − 169.·22-s + 23·23-s + 22.7·24-s + 260.·25-s + 84.0·26-s + 27·27-s + 68.5·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.22·4-s − 1.75·5-s + 0.861·6-s + 0.377·7-s + 0.335·8-s + 0.333·9-s − 2.61·10-s − 1.10·11-s + 0.707·12-s + 0.424·13-s + 0.563·14-s − 1.01·15-s − 0.724·16-s − 1.38·17-s + 0.497·18-s − 1.49·19-s − 2.14·20-s + 0.218·21-s − 1.64·22-s + 0.208·23-s + 0.193·24-s + 2.08·25-s + 0.633·26-s + 0.192·27-s + 0.462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.21T + 8T^{2} \) |
| 5 | \( 1 + 19.6T + 125T^{2} \) |
| 11 | \( 1 + 40.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 124.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 89.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 226.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 93.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 303.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 503.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 875.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 255.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 715.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 996.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 440.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.56e3T + 9.12e5T^{2} \) |
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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
−10.72425539981009308043617488138, −8.825724826727341031356285076528, −8.239556978612378041468170698727, −7.28118457628807093736128793815, −6.33909968083670030857493036230, −4.80430913754539419142088088002, −4.34748449954259047571441722062, −3.40782918821296401879796381276, −2.38533308583294802717119625484, 0,
2.38533308583294802717119625484, 3.40782918821296401879796381276, 4.34748449954259047571441722062, 4.80430913754539419142088088002, 6.33909968083670030857493036230, 7.28118457628807093736128793815, 8.239556978612378041468170698727, 8.825724826727341031356285076528, 10.72425539981009308043617488138
