| L(s) = 1 | − 2-s + 4-s − 8-s − 11-s + 2·13-s + 16-s + 6·17-s + 4·19-s + 22-s − 4·23-s − 2·26-s − 2·29-s + 4·31-s − 32-s − 6·34-s + 2·37-s − 4·38-s − 6·41-s − 44-s + 4·46-s + 8·47-s + 2·52-s − 14·53-s + 2·58-s + 12·59-s + 14·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.301·11-s + 0.554·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.213·22-s − 0.834·23-s − 0.392·26-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.328·37-s − 0.648·38-s − 0.937·41-s − 0.150·44-s + 0.589·46-s + 1.16·47-s + 0.277·52-s − 1.92·53-s + 0.262·58-s + 1.56·59-s + 1.79·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97835964500525, −12.64014421050213, −12.04593259896011, −11.60334614327201, −11.39232824749663, −10.66752297768615, −10.16585933589125, −9.938205684533950, −9.493849602248778, −8.865051115075412, −8.397952371032575, −7.964650204753652, −7.556308788161085, −7.136896539657106, −6.427217908255064, −6.056203247868233, −5.382317502784713, −5.201661429335953, −4.266148003179156, −3.732207775907045, −3.211628019456352, −2.691473347330854, −1.999849451338099, −1.298045362405483, −0.8769926648720510, 0,
0.8769926648720510, 1.298045362405483, 1.999849451338099, 2.691473347330854, 3.211628019456352, 3.732207775907045, 4.266148003179156, 5.201661429335953, 5.382317502784713, 6.056203247868233, 6.427217908255064, 7.136896539657106, 7.556308788161085, 7.964650204753652, 8.397952371032575, 8.865051115075412, 9.493849602248778, 9.938205684533950, 10.16585933589125, 10.66752297768615, 11.39232824749663, 11.60334614327201, 12.04593259896011, 12.64014421050213, 12.97835964500525
