| L(s) = 1 | − 2.61·2-s − 3·3-s − 1.14·4-s − 7.96·5-s + 7.85·6-s + 16.8·7-s + 23.9·8-s + 9·9-s + 20.8·10-s + 62.7·11-s + 3.42·12-s − 75.2·13-s − 44.1·14-s + 23.9·15-s − 53.5·16-s − 51.9·17-s − 23.5·18-s + 24.2·19-s + 9.09·20-s − 50.6·21-s − 164.·22-s − 9.85·23-s − 71.8·24-s − 61.5·25-s + 196.·26-s − 27·27-s − 19.2·28-s + ⋯ |
| L(s) = 1 | − 0.925·2-s − 0.577·3-s − 0.142·4-s − 0.712·5-s + 0.534·6-s + 0.911·7-s + 1.05·8-s + 0.333·9-s + 0.659·10-s + 1.72·11-s + 0.0823·12-s − 1.60·13-s − 0.843·14-s + 0.411·15-s − 0.836·16-s − 0.741·17-s − 0.308·18-s + 0.292·19-s + 0.101·20-s − 0.526·21-s − 1.59·22-s − 0.0893·23-s − 0.610·24-s − 0.492·25-s + 1.48·26-s − 0.192·27-s − 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 + 2.61T + 8T^{2} \) |
| 5 | \( 1 + 7.96T + 125T^{2} \) |
| 7 | \( 1 - 16.8T + 343T^{2} \) |
| 11 | \( 1 - 62.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 75.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.85T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 125.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 467.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 90.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 437.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 353.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 697.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 292.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 223.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 122.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 662.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 397.T + 9.12e5T^{2} \) |
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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
−11.71161811450465143587196369629, −10.71144461773026984886634169203, −9.631672300621368549961105600445, −8.744507303002251351049037388776, −7.67095305676505166620895010483, −6.82151572149032375375175816024, −4.98330391686472106529166112028, −4.13406633749339021797690890030, −1.57054288843129401275024617155, 0,
1.57054288843129401275024617155, 4.13406633749339021797690890030, 4.98330391686472106529166112028, 6.82151572149032375375175816024, 7.67095305676505166620895010483, 8.744507303002251351049037388776, 9.631672300621368549961105600445, 10.71144461773026984886634169203, 11.71161811450465143587196369629
