| L(s) = 1 | − 2.48·3-s + 2.48·5-s + 7-s + 3.15·9-s − 1.19·11-s − 2.96·13-s − 6.15·15-s − 0.481·17-s + 4.31·19-s − 2.48·21-s − 23-s + 1.15·25-s − 0.387·27-s − 4.15·29-s − 10.2·31-s + 2.96·33-s + 2.48·35-s − 5.35·37-s + 7.35·39-s − 11.9·41-s + 5.73·43-s + 7.83·45-s − 0.168·47-s + 49-s + 1.19·51-s + 8.70·53-s − 2.96·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 1.10·5-s + 0.377·7-s + 1.05·9-s − 0.359·11-s − 0.821·13-s − 1.58·15-s − 0.116·17-s + 0.989·19-s − 0.541·21-s − 0.208·23-s + 0.231·25-s − 0.0746·27-s − 0.771·29-s − 1.83·31-s + 0.515·33-s + 0.419·35-s − 0.879·37-s + 1.17·39-s − 1.86·41-s + 0.875·43-s + 1.16·45-s − 0.0245·47-s + 0.142·49-s + 0.167·51-s + 1.19·53-s − 0.399·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 11 | \( 1 + 1.19T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 0.481T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 29 | \( 1 + 4.15T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 5.73T + 43T^{2} \) |
| 47 | \( 1 + 0.168T + 47T^{2} \) |
| 53 | \( 1 - 8.70T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 9.05T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 - 8.88T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 13.7T + 89T^{2} \) |
| 97 | \( 1 + 0.0933T + 97T^{2} \) |
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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
−8.620002261208970802779556517951, −7.40287245225550919884950618077, −6.95521748894530564596448469893, −5.91608093094720594645770313367, −5.36493265033466108324758502485, −5.06415526124824666278760103570, −3.78791003738275179145296236210, −2.39042729166671274995334181525, −1.45373004443178733836707682807, 0,
1.45373004443178733836707682807, 2.39042729166671274995334181525, 3.78791003738275179145296236210, 5.06415526124824666278760103570, 5.36493265033466108324758502485, 5.91608093094720594645770313367, 6.95521748894530564596448469893, 7.40287245225550919884950618077, 8.620002261208970802779556517951
