| L(s) = 1 | + 3·7-s − 5·11-s + 13-s − 17-s + 19-s − 7·29-s − 10·31-s + 3·37-s − 9·41-s − 8·43-s − 3·47-s + 2·49-s − 53-s − 4·59-s − 10·61-s − 6·67-s − 8·71-s + 13·73-s − 15·77-s − 10·79-s + 12·83-s − 14·89-s + 3·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 1.50·11-s + 0.277·13-s − 0.242·17-s + 0.229·19-s − 1.29·29-s − 1.79·31-s + 0.493·37-s − 1.40·41-s − 1.21·43-s − 0.437·47-s + 2/7·49-s − 0.137·53-s − 0.520·59-s − 1.28·61-s − 0.733·67-s − 0.949·71-s + 1.52·73-s − 1.70·77-s − 1.12·79-s + 1.31·83-s − 1.48·89-s + 0.314·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.08014870409921, −12.42351246474885, −11.98325651241018, −11.27024156575973, −11.23389709350387, −10.66274158701660, −10.37131123303855, −9.677319770076773, −9.332217619026522, −8.685216275298423, −8.333333820238010, −7.827388541119434, −7.549832377877560, −7.075988092787098, −6.453678981204423, −5.803154880127767, −5.358384997014080, −5.071865103979946, −4.585495529013932, −3.955119616169411, −3.361616256674588, −2.909664505747371, −2.151057744818356, −1.734967562725886, −1.283942744387718, 0, 0,
1.283942744387718, 1.734967562725886, 2.151057744818356, 2.909664505747371, 3.361616256674588, 3.955119616169411, 4.585495529013932, 5.071865103979946, 5.358384997014080, 5.803154880127767, 6.453678981204423, 7.075988092787098, 7.549832377877560, 7.827388541119434, 8.333333820238010, 8.685216275298423, 9.332217619026522, 9.677319770076773, 10.37131123303855, 10.66274158701660, 11.23389709350387, 11.27024156575973, 11.98325651241018, 12.42351246474885, 13.08014870409921
