| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s + 16-s + 6·17-s − 4·19-s + 20-s + 4·22-s + 23-s + 25-s + 2·29-s + 32-s + 6·34-s + 2·37-s − 4·38-s + 40-s + 10·41-s − 4·43-s + 4·44-s + 46-s − 7·49-s + 50-s − 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.371·29-s + 0.176·32-s + 1.02·34-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s + 0.147·46-s − 49-s + 0.141·50-s − 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.71424268965168, −12.49994302205694, −11.88152942892547, −11.49088449800047, −11.13527905555136, −10.44193423102630, −10.18424016351013, −9.596004038502540, −9.200738663888432, −8.709248277715157, −8.150417630697633, −7.574503774257557, −7.261547161326523, −6.503830471755341, −6.144924785107276, −5.999873487192241, −5.179177879618700, −4.789796847952658, −4.257002556630372, −3.732811916655277, −3.242983630828381, −2.714078570850892, −2.089385534716637, −1.333976646116673, −1.138103075642842, 0,
1.138103075642842, 1.333976646116673, 2.089385534716637, 2.714078570850892, 3.242983630828381, 3.732811916655277, 4.257002556630372, 4.789796847952658, 5.179177879618700, 5.999873487192241, 6.144924785107276, 6.503830471755341, 7.261547161326523, 7.574503774257557, 8.150417630697633, 8.709248277715157, 9.200738663888432, 9.596004038502540, 10.18424016351013, 10.44193423102630, 11.13527905555136, 11.49088449800047, 11.88152942892547, 12.49994302205694, 12.71424268965168
