| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 18.4·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 36.8·10-s − 50.2·11-s − 12·12-s + 85.6·13-s + 14·14-s + 55.3·15-s + 16·16-s − 92.4·17-s − 18·18-s + 9.31·19-s − 73.7·20-s + 21·21-s + 100.·22-s + 23·23-s + 24·24-s + 215.·25-s − 171.·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.64·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s − 1.37·11-s − 0.288·12-s + 1.82·13-s + 0.267·14-s + 0.952·15-s + 0.250·16-s − 1.31·17-s − 0.235·18-s + 0.112·19-s − 0.824·20-s + 0.218·21-s + 0.973·22-s + 0.208·23-s + 0.204·24-s + 1.72·25-s − 1.29·26-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 + 18.4T + 125T^{2} \) |
| 11 | \( 1 + 50.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 85.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.31T + 6.85e3T^{2} \) |
| 29 | \( 1 - 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 90.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 501.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 108.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 227.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 941.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 525.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 75.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 974.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 517.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.28e3T + 9.12e5T^{2} \) |
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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
−8.974458520875372682386639792103, −8.358699557685867581968578567753, −7.67056547369659192235570627724, −6.80124507705021708422323703943, −5.98611373666064363874901627442, −4.70246556094612531613182618492, −3.79698423103443998816551898522, −2.71749309113586332582390708313, −0.923642022706226285155562646246, 0,
0.923642022706226285155562646246, 2.71749309113586332582390708313, 3.79698423103443998816551898522, 4.70246556094612531613182618492, 5.98611373666064363874901627442, 6.80124507705021708422323703943, 7.67056547369659192235570627724, 8.358699557685867581968578567753, 8.974458520875372682386639792103
