| L(s) = 1 | + 0.462·3-s − 5-s + 2.86·7-s − 2.78·9-s − 5.72·11-s + 0.462·13-s − 0.462·15-s + 1.93·17-s + 8.36·19-s + 1.32·21-s − 8.18·23-s + 25-s − 2.67·27-s − 29-s − 3.78·31-s − 2.64·33-s − 2.86·35-s + 3.07·37-s + 0.213·39-s + 3.72·41-s − 9.78·43-s + 2.78·45-s − 8.64·47-s + 1.18·49-s + 0.895·51-s − 2.58·53-s + 5.72·55-s + ⋯ |
| L(s) = 1 | + 0.267·3-s − 0.447·5-s + 1.08·7-s − 0.928·9-s − 1.72·11-s + 0.128·13-s − 0.119·15-s + 0.469·17-s + 1.91·19-s + 0.288·21-s − 1.70·23-s + 0.200·25-s − 0.515·27-s − 0.185·29-s − 0.679·31-s − 0.460·33-s − 0.483·35-s + 0.505·37-s + 0.0342·39-s + 0.581·41-s − 1.49·43-s + 0.415·45-s − 1.26·47-s + 0.169·49-s + 0.125·51-s − 0.354·53-s + 0.771·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.462T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 + 5.72T + 11T^{2} \) |
| 13 | \( 1 - 0.462T + 13T^{2} \) |
| 17 | \( 1 - 1.93T + 17T^{2} \) |
| 19 | \( 1 - 8.36T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 31 | \( 1 + 3.78T + 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 - 5.90T + 79T^{2} \) |
| 83 | \( 1 + 4.92T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.315096360812951787817330946009, −7.79185177576285762709687639765, −7.61446426977454530957297556422, −6.08251934172827433262852935396, −5.32873417586353388178986597043, −4.80120928138482363700496745374, −3.51457714571073270727550499443, −2.79939233125817263254683959022, −1.65885597051326279955354370982, 0,
1.65885597051326279955354370982, 2.79939233125817263254683959022, 3.51457714571073270727550499443, 4.80120928138482363700496745374, 5.32873417586353388178986597043, 6.08251934172827433262852935396, 7.61446426977454530957297556422, 7.79185177576285762709687639765, 8.315096360812951787817330946009
