| L(s) = 1 | + 3-s − 5.12·7-s + 9-s − 2·11-s + 5.12·13-s + 1.12·17-s − 5.12·19-s − 5.12·21-s + 5.12·23-s + 27-s + 8.24·29-s − 7.12·31-s − 2·33-s − 5.12·37-s + 5.12·39-s − 2·41-s − 6.24·43-s + 13.1·47-s + 19.2·49-s + 1.12·51-s + 10·53-s − 5.12·57-s − 6·59-s + 2·61-s − 5.12·63-s − 6.24·67-s + 5.12·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.93·7-s + 0.333·9-s − 0.603·11-s + 1.42·13-s + 0.272·17-s − 1.17·19-s − 1.11·21-s + 1.06·23-s + 0.192·27-s + 1.53·29-s − 1.27·31-s − 0.348·33-s − 0.842·37-s + 0.820·39-s − 0.312·41-s − 0.952·43-s + 1.91·47-s + 2.74·49-s + 0.157·51-s + 1.37·53-s − 0.678·57-s − 0.781·59-s + 0.256·61-s − 0.645·63-s − 0.763·67-s + 0.616·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.09537559799055311537090452770, −6.81882686798951712409911830636, −6.03914864632514874173734439037, −5.48167904606194848887664689255, −4.32049754171634515435376014926, −3.64670035272156593599720885924, −3.08218791960758770941230954405, −2.43573341256551569215538879444, −1.18560905016200391312150246310, 0,
1.18560905016200391312150246310, 2.43573341256551569215538879444, 3.08218791960758770941230954405, 3.64670035272156593599720885924, 4.32049754171634515435376014926, 5.48167904606194848887664689255, 6.03914864632514874173734439037, 6.81882686798951712409911830636, 7.09537559799055311537090452770
