| L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s − 3·11-s + 13-s + 14-s + 16-s + 3·17-s − 7·19-s − 3·20-s − 3·22-s − 9·23-s + 4·25-s + 26-s + 28-s + 9·29-s − 4·31-s + 32-s + 3·34-s − 3·35-s − 7·37-s − 7·38-s − 3·40-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.670·20-s − 0.639·22-s − 1.87·23-s + 4/5·25-s + 0.196·26-s + 0.188·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 0.507·35-s − 1.15·37-s − 1.13·38-s − 0.474·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.494657990807389898658506219562, −8.231691217522472057103385997568, −7.42970330921915709586726435368, −6.54796581741576703271548356667, −5.59399109559353101448430855804, −4.65012025043354629223454819364, −3.98883124750352014007177515851, −3.13495261897234347757421228131, −1.89338424078267539825922227519, 0,
1.89338424078267539825922227519, 3.13495261897234347757421228131, 3.98883124750352014007177515851, 4.65012025043354629223454819364, 5.59399109559353101448430855804, 6.54796581741576703271548356667, 7.42970330921915709586726435368, 8.231691217522472057103385997568, 8.494657990807389898658506219562
