| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5.36·5-s + 6·6-s + 7·7-s + 8·8-s + 9·9-s − 10.7·10-s − 14.6·11-s + 12·12-s + 64.7·13-s + 14·14-s − 16.0·15-s + 16·16-s + 24.9·17-s + 18·18-s − 5.71·19-s − 21.4·20-s + 21·21-s − 29.2·22-s + 23·23-s + 24·24-s − 96.2·25-s + 129.·26-s + 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.479·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.339·10-s − 0.401·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s − 0.276·15-s + 0.250·16-s + 0.355·17-s + 0.235·18-s − 0.0690·19-s − 0.239·20-s + 0.218·21-s − 0.283·22-s + 0.208·23-s + 0.204·24-s − 0.769·25-s + 0.977·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.346024641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.346024641\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 5.36T + 125T^{2} \) |
| 11 | \( 1 + 14.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.71T + 6.85e3T^{2} \) |
| 29 | \( 1 - 136.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 90.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 400.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 214.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 460.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 307.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 730.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 554.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 956.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 360.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 300.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 703.T + 9.12e5T^{2} \) |
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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
−9.686706340141334977532561097246, −8.476001169243313517027672592430, −8.088302448544275265339010142108, −7.09542504042921736149025543217, −6.15373600300984777910489539166, −5.17263487585467408290343964058, −4.13364595924487138667082713342, −3.44420251936257816391806685866, −2.33266168614738398006703694195, −1.04024050043123796864273184725,
1.04024050043123796864273184725, 2.33266168614738398006703694195, 3.44420251936257816391806685866, 4.13364595924487138667082713342, 5.17263487585467408290343964058, 6.15373600300984777910489539166, 7.09542504042921736149025543217, 8.088302448544275265339010142108, 8.476001169243313517027672592430, 9.686706340141334977532561097246
