| L(s) = 1 | + 2-s − 2·3-s + 4-s + 2·5-s − 2·6-s + 8-s + 9-s + 2·10-s − 2·12-s − 4·13-s − 4·15-s + 16-s − 4·17-s + 18-s − 8·19-s + 2·20-s − 4·23-s − 2·24-s − 25-s − 4·26-s + 4·27-s + 2·29-s − 4·30-s − 31-s + 32-s − 4·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.577·12-s − 1.10·13-s − 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.834·23-s − 0.408·24-s − 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.371·29-s − 0.730·30-s − 0.179·31-s + 0.176·32-s − 0.685·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321346 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321346 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.85125017396340, −12.69100254951639, −12.25198662252156, −11.65954344086348, −11.43285717223779, −10.76592774153722, −10.52035502218142, −10.03633383658503, −9.650016183729629, −8.972609636875429, −8.474171595452268, −7.999203589444829, −7.241721941655390, −6.761064500616634, −6.439321526414087, −6.026238859671002, −5.592795758630321, −5.098242176862005, −4.630189130855104, −4.262268833762220, −3.639514333753447, −2.762010817152897, −2.255939718673929, −1.974975096331478, −1.177927986793116, 0, 0,
1.177927986793116, 1.974975096331478, 2.255939718673929, 2.762010817152897, 3.639514333753447, 4.262268833762220, 4.630189130855104, 5.098242176862005, 5.592795758630321, 6.026238859671002, 6.439321526414087, 6.761064500616634, 7.241721941655390, 7.999203589444829, 8.474171595452268, 8.972609636875429, 9.650016183729629, 10.03633383658503, 10.52035502218142, 10.76592774153722, 11.43285717223779, 11.65954344086348, 12.25198662252156, 12.69100254951639, 12.85125017396340
