| L(s) = 1 | + 2·5-s − 7-s + 2·17-s − 4·19-s + 4·23-s − 25-s + 10·29-s + 8·31-s − 2·35-s − 6·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 10·53-s + 12·59-s − 2·61-s − 12·67-s − 12·71-s + 14·73-s − 8·79-s + 12·83-s + 4·85-s − 2·89-s − 8·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s − 0.256·61-s − 1.46·67-s − 1.42·71-s + 1.63·73-s − 0.900·79-s + 1.31·83-s + 0.433·85-s − 0.211·89-s − 0.820·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.069927531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.069927531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.82383706873045, −13.46224296435876, −13.10501838353834, −12.38123083219462, −12.00288983241015, −11.62629651839277, −10.66647358701882, −10.37832021012591, −10.07719937299980, −9.481048096942856, −8.814174257379094, −8.552913703291568, −7.921265566089268, −7.183874523070328, −6.575600037216498, −6.404758484666947, −5.592442673839928, −5.253589495858352, −4.495798838470012, −4.009130282971311, −3.127977723967246, −2.691162316681836, −2.062276382636163, −1.268815905823931, −0.5929186627260467,
0.5929186627260467, 1.268815905823931, 2.062276382636163, 2.691162316681836, 3.127977723967246, 4.009130282971311, 4.495798838470012, 5.253589495858352, 5.592442673839928, 6.404758484666947, 6.575600037216498, 7.183874523070328, 7.921265566089268, 8.552913703291568, 8.814174257379094, 9.481048096942856, 10.07719937299980, 10.37832021012591, 10.66647358701882, 11.62629651839277, 12.00288983241015, 12.38123083219462, 13.10501838353834, 13.46224296435876, 13.82383706873045
