| L(s) = 1 | − 1.79·3-s + 3.68·5-s + 7-s + 0.223·9-s + 11-s − 13-s − 6.60·15-s − 5.37·17-s − 1.25·19-s − 1.79·21-s + 1.60·23-s + 8.54·25-s + 4.98·27-s − 4.82·29-s − 2.42·31-s − 1.79·33-s + 3.68·35-s + 4.86·37-s + 1.79·39-s − 12.3·41-s − 2.12·43-s + 0.824·45-s + 10.1·47-s + 49-s + 9.65·51-s − 2.26·53-s + 3.68·55-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 1.64·5-s + 0.377·7-s + 0.0746·9-s + 0.301·11-s − 0.277·13-s − 1.70·15-s − 1.30·17-s − 0.288·19-s − 0.391·21-s + 0.335·23-s + 1.70·25-s + 0.959·27-s − 0.896·29-s − 0.436·31-s − 0.312·33-s + 0.622·35-s + 0.799·37-s + 0.287·39-s − 1.92·41-s − 0.324·43-s + 0.122·45-s + 1.47·47-s + 0.142·49-s + 1.35·51-s − 0.311·53-s + 0.496·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 + 4.82T + 29T^{2} \) |
| 31 | \( 1 + 2.42T + 31T^{2} \) |
| 37 | \( 1 - 4.86T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 + 0.165T + 67T^{2} \) |
| 71 | \( 1 + 4.36T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
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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
−7.17996683160276698865064237452, −6.58372565025260161110512093067, −6.08292449388335834425745372038, −5.43693798008114562159720559639, −4.96008519442480126760728201734, −4.16386266087062649086922807975, −2.87123695096230459548804518693, −2.07328041605003480890090777344, −1.34391270135139237027720127639, 0,
1.34391270135139237027720127639, 2.07328041605003480890090777344, 2.87123695096230459548804518693, 4.16386266087062649086922807975, 4.96008519442480126760728201734, 5.43693798008114562159720559639, 6.08292449388335834425745372038, 6.58372565025260161110512093067, 7.17996683160276698865064237452
