| L(s) = 1 | + 3·3-s + 2·5-s + 7-s + 6·9-s − 2·11-s + 3·13-s + 6·15-s − 8·19-s + 3·21-s + 4·23-s − 25-s + 9·27-s − 10·29-s + 6·31-s − 6·33-s + 2·35-s − 2·37-s + 9·39-s − 4·43-s + 12·45-s − 12·47-s + 49-s + 12·53-s − 4·55-s − 24·57-s − 11·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 0.377·7-s + 2·9-s − 0.603·11-s + 0.832·13-s + 1.54·15-s − 1.83·19-s + 0.654·21-s + 0.834·23-s − 1/5·25-s + 1.73·27-s − 1.85·29-s + 1.07·31-s − 1.04·33-s + 0.338·35-s − 0.328·37-s + 1.44·39-s − 0.609·43-s + 1.78·45-s − 1.75·47-s + 1/7·49-s + 1.64·53-s − 0.539·55-s − 3.17·57-s − 1.43·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.31146262469478, −13.18297185202351, −12.74387297592937, −12.10066374044154, −11.32491822331520, −10.84750386014454, −10.47524899598328, −9.887377339045174, −9.525697556360101, −8.974036519224172, −8.616364214702515, −8.227735527514794, −7.765084805312716, −7.272653497239674, −6.524530689069010, −6.270235136126179, −5.503712056242022, −4.932960365130983, −4.378074419514985, −3.725945938544281, −3.382137964589774, −2.613437901536700, −2.186200624423737, −1.774489157211496, −1.201113382519642, 0,
1.201113382519642, 1.774489157211496, 2.186200624423737, 2.613437901536700, 3.382137964589774, 3.725945938544281, 4.378074419514985, 4.932960365130983, 5.503712056242022, 6.270235136126179, 6.524530689069010, 7.272653497239674, 7.765084805312716, 8.227735527514794, 8.616364214702515, 8.974036519224172, 9.525697556360101, 9.887377339045174, 10.47524899598328, 10.84750386014454, 11.32491822331520, 12.10066374044154, 12.74387297592937, 13.18297185202351, 13.31146262469478
