L(s) = 1 | − 9.70·2-s + 9·3-s + 62.1·4-s − 87.3·6-s + 79.4·7-s − 292.·8-s + 81·9-s − 121·11-s + 559.·12-s + 780.·13-s − 770.·14-s + 848.·16-s − 1.97e3·17-s − 785.·18-s + 2.31e3·19-s + 714.·21-s + 1.17e3·22-s − 1.28e3·23-s − 2.63e3·24-s − 7.57e3·26-s + 729·27-s + 4.93e3·28-s − 1.27e3·29-s − 6.82e3·31-s + 1.12e3·32-s − 1.08e3·33-s + 1.91e4·34-s + ⋯ |
L(s) = 1 | − 1.71·2-s + 0.577·3-s + 1.94·4-s − 0.990·6-s + 0.612·7-s − 1.61·8-s + 0.333·9-s − 0.301·11-s + 1.12·12-s + 1.28·13-s − 1.05·14-s + 0.828·16-s − 1.65·17-s − 0.571·18-s + 1.47·19-s + 0.353·21-s + 0.517·22-s − 0.505·23-s − 0.932·24-s − 2.19·26-s + 0.192·27-s + 1.18·28-s − 0.282·29-s − 1.27·31-s + 0.193·32-s − 0.174·33-s + 2.84·34-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 | \( 1 + 9.70T + 32T^{2} \) |
| 7 | \( 1 - 79.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 780.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.47e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.33e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.36e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.50e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.01e4T + 8.58e9T^{2} \) |
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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
−8.999214432622135375396665832217, −8.346571042932911759380098705280, −7.68337804819737272872073018255, −6.90468621355303318896445727132, −5.86224724079068199212680755748, −4.43885809451466348813120656603, −3.13330553160019967231035485165, −1.97607257947709297115868743959, −1.26331695637334938689483274012, 0,
1.26331695637334938689483274012, 1.97607257947709297115868743959, 3.13330553160019967231035485165, 4.43885809451466348813120656603, 5.86224724079068199212680755748, 6.90468621355303318896445727132, 7.68337804819737272872073018255, 8.346571042932911759380098705280, 8.999214432622135375396665832217