| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 21-s − 22-s − 24-s + 25-s − 2·26-s + 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−15.03071548735733, −14.60678385180776, −14.03302348715042, −13.66776912842595, −12.86816790965363, −12.61215412707311, −11.84564383063097, −11.22585278057930, −10.84167258728862, −10.18046277783600, −9.825229204955544, −9.069439468972885, −8.675664545943381, −8.356273490951712, −7.616055175972977, −7.075028278125713, −6.452754048257517, −5.977573346815168, −5.239868859950243, −4.397202309275230, −3.944661945057742, −2.986778040185100, −2.505513967121129, −1.654554382891277, −1.221565515433384, 0,
1.221565515433384, 1.654554382891277, 2.505513967121129, 2.986778040185100, 3.944661945057742, 4.397202309275230, 5.239868859950243, 5.977573346815168, 6.452754048257517, 7.075028278125713, 7.616055175972977, 8.356273490951712, 8.675664545943381, 9.069439468972885, 9.825229204955544, 10.18046277783600, 10.84167258728862, 11.22585278057930, 11.84564383063097, 12.61215412707311, 12.86816790965363, 13.66776912842595, 14.03302348715042, 14.60678385180776, 15.03071548735733
