| L(s) = 1 | − 0.456·2-s − 1.79·4-s + 0.791·7-s + 1.73·8-s − 2.18·11-s − 1.20·13-s − 0.361·14-s + 2.79·16-s + 5.74·17-s − 3.58·19-s + 0.999·22-s − 8.29·23-s + 0.552·26-s − 1.41·28-s + 5.29·29-s + 3·31-s − 4.73·32-s − 2.62·34-s − 0.208·37-s + 1.63·38-s + 9.11·41-s − 10.3·43-s + 3.92·44-s + 3.79·46-s − 0.818·47-s − 6.37·49-s + 2.16·52-s + ⋯ |
| L(s) = 1 | − 0.323·2-s − 0.895·4-s + 0.299·7-s + 0.612·8-s − 0.659·11-s − 0.335·13-s − 0.0966·14-s + 0.697·16-s + 1.39·17-s − 0.821·19-s + 0.213·22-s − 1.73·23-s + 0.108·26-s − 0.267·28-s + 0.982·29-s + 0.538·31-s − 0.837·32-s − 0.450·34-s − 0.0343·37-s + 0.265·38-s + 1.42·41-s − 1.58·43-s + 0.591·44-s + 0.558·46-s − 0.119·47-s − 0.910·49-s + 0.300·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.456T + 2T^{2} \) |
| 7 | \( 1 - 0.791T + 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 - 5.74T + 17T^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 23 | \( 1 + 8.29T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 0.208T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 0.818T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 6.37T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8.85T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 5.19T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
| 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
−8.628197691256777203834237865306, −8.036570272695106157547145301744, −7.56703521591005269938379997611, −6.29387992379362953122056532951, −5.47303063310723579680606277952, −4.67646736068232938364622042275, −3.91235544185783026917682142860, −2.74666020080243266589755569770, −1.42392998387350866834844827431, 0,
1.42392998387350866834844827431, 2.74666020080243266589755569770, 3.91235544185783026917682142860, 4.67646736068232938364622042275, 5.47303063310723579680606277952, 6.29387992379362953122056532951, 7.56703521591005269938379997611, 8.036570272695106157547145301744, 8.628197691256777203834237865306
