| L(s) = 1 | − 3-s − 7-s + 9-s + 5.65·11-s − 2·13-s + 3.65·17-s − 5.65·19-s + 21-s + 5.65·23-s − 27-s + 3.65·29-s + 4·31-s − 5.65·33-s − 11.6·37-s + 2·39-s + 2·41-s − 1.65·43-s + 2.34·47-s + 49-s − 3.65·51-s + 3.65·53-s + 5.65·57-s − 4·59-s + 0.343·61-s − 63-s + 9.65·67-s − 5.65·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 0.333·9-s + 1.70·11-s − 0.554·13-s + 0.886·17-s − 1.29·19-s + 0.218·21-s + 1.17·23-s − 0.192·27-s + 0.679·29-s + 0.718·31-s − 0.984·33-s − 1.91·37-s + 0.320·39-s + 0.312·41-s − 0.252·43-s + 0.341·47-s + 0.142·49-s − 0.512·51-s + 0.502·53-s + 0.749·57-s − 0.520·59-s + 0.0439·61-s − 0.125·63-s + 1.17·67-s − 0.681·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580741640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580741640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 - 2.34T + 47T^{2} \) |
| 53 | \( 1 - 3.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 0.343T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.601391745108637722899934915740, −7.49267321886468964500898938259, −6.73062210066174641407530229062, −6.40613991462151765412235079546, −5.46569926755382447838056336673, −4.64889618335368037927417033431, −3.90126385717873024788387985266, −3.03854100236757681317097516293, −1.78487256949889422398927118715, −0.75955387726228538778460445762,
0.75955387726228538778460445762, 1.78487256949889422398927118715, 3.03854100236757681317097516293, 3.90126385717873024788387985266, 4.64889618335368037927417033431, 5.46569926755382447838056336673, 6.40613991462151765412235079546, 6.73062210066174641407530229062, 7.49267321886468964500898938259, 8.601391745108637722899934915740
