| L(s) = 1 | − 2.60·2-s + 9·3-s − 25.1·4-s − 23.4·6-s + 174.·7-s + 149.·8-s + 81·9-s + 121·11-s − 226.·12-s − 815.·13-s − 456.·14-s + 416.·16-s + 462.·17-s − 211.·18-s − 1.99e3·19-s + 1.57e3·21-s − 315.·22-s + 1.34e3·23-s + 1.34e3·24-s + 2.12e3·26-s + 729·27-s − 4.40e3·28-s + 6.58e3·29-s − 4.49e3·31-s − 5.86e3·32-s + 1.08e3·33-s − 1.20e3·34-s + ⋯ |
| L(s) = 1 | − 0.461·2-s + 0.577·3-s − 0.787·4-s − 0.266·6-s + 1.34·7-s + 0.824·8-s + 0.333·9-s + 0.301·11-s − 0.454·12-s − 1.33·13-s − 0.622·14-s + 0.407·16-s + 0.388·17-s − 0.153·18-s − 1.26·19-s + 0.779·21-s − 0.139·22-s + 0.529·23-s + 0.475·24-s + 0.617·26-s + 0.192·27-s − 1.06·28-s + 1.45·29-s − 0.839·31-s − 1.01·32-s + 0.174·33-s − 0.179·34-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.60T + 32T^{2} \) |
| 7 | \( 1 - 174.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 815.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 462.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.49e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.09e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.93e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.12e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.19e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.80e5T + 8.58e9T^{2} \) |
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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
−8.856451858854642394372428771086, −8.340891973979042317580053135979, −7.67029717082913828509095295466, −6.72512598825535730022664541834, −5.04475317920928906263383441448, −4.76019867506817493026901997372, −3.60905552699983414160064663086, −2.19342892276920978944989755982, −1.30801600907136062872419968018, 0,
1.30801600907136062872419968018, 2.19342892276920978944989755982, 3.60905552699983414160064663086, 4.76019867506817493026901997372, 5.04475317920928906263383441448, 6.72512598825535730022664541834, 7.67029717082913828509095295466, 8.340891973979042317580053135979, 8.856451858854642394372428771086
