| L(s) = 1 | + 0.624·2-s + 9·3-s − 31.6·4-s + 5.61·6-s + 106.·7-s − 39.7·8-s + 81·9-s − 121·11-s − 284.·12-s + 264.·13-s + 66.3·14-s + 986.·16-s + 509.·17-s + 50.5·18-s + 1.66e3·19-s + 956.·21-s − 75.5·22-s + 2.21e3·23-s − 357.·24-s + 165.·26-s + 729·27-s − 3.35e3·28-s − 5.42e3·29-s − 8.79e3·31-s + 1.88e3·32-s − 1.08e3·33-s + 318.·34-s + ⋯ |
| L(s) = 1 | + 0.110·2-s + 0.577·3-s − 0.987·4-s + 0.0637·6-s + 0.819·7-s − 0.219·8-s + 0.333·9-s − 0.301·11-s − 0.570·12-s + 0.433·13-s + 0.0904·14-s + 0.963·16-s + 0.427·17-s + 0.0367·18-s + 1.05·19-s + 0.473·21-s − 0.0332·22-s + 0.871·23-s − 0.126·24-s + 0.0478·26-s + 0.192·27-s − 0.809·28-s − 1.19·29-s − 1.64·31-s + 0.325·32-s − 0.174·33-s + 0.0471·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.645854623\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645854623\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 0.624T + 32T^{2} \) |
| 7 | \( 1 - 106.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 264.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 509.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.21e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.42e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.52e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.46e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.18e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.95e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.38e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.89e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.60e4T + 8.58e9T^{2} \) |
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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
−9.262470732372142190757814877877, −8.741337714448484245716262597973, −7.84872843238503684417862029813, −7.20861427346647069491537190165, −5.60650602281839597794580175488, −5.09553479592040431045160588080, −3.96804944214712236918888294163, −3.22882570733220323277235341457, −1.80121942258648731382952959501, −0.74013548805860653409634626621,
0.74013548805860653409634626621, 1.80121942258648731382952959501, 3.22882570733220323277235341457, 3.96804944214712236918888294163, 5.09553479592040431045160588080, 5.60650602281839597794580175488, 7.20861427346647069491537190165, 7.84872843238503684417862029813, 8.741337714448484245716262597973, 9.262470732372142190757814877877
