| L(s) = 1 | − 1.05·3-s − 1.23·5-s + 7-s − 1.87·9-s − 1.37·11-s + 1.31·15-s + 1.41·17-s − 6.61·19-s − 1.05·21-s − 5.86·23-s − 3.46·25-s + 5.16·27-s − 5.41·29-s − 3.88·31-s + 1.45·33-s − 1.23·35-s + 0.373·37-s − 1.42·41-s − 5.83·43-s + 2.32·45-s − 0.595·47-s + 49-s − 1.50·51-s + 4.15·53-s + 1.69·55-s + 7.01·57-s + 0.154·59-s + ⋯ |
| L(s) = 1 | − 0.611·3-s − 0.553·5-s + 0.377·7-s − 0.626·9-s − 0.413·11-s + 0.338·15-s + 0.344·17-s − 1.51·19-s − 0.231·21-s − 1.22·23-s − 0.693·25-s + 0.994·27-s − 1.00·29-s − 0.697·31-s + 0.252·33-s − 0.209·35-s + 0.0614·37-s − 0.223·41-s − 0.890·43-s + 0.346·45-s − 0.0868·47-s + 0.142·49-s − 0.210·51-s + 0.571·53-s + 0.228·55-s + 0.928·57-s + 0.0201·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4671501961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4671501961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 - 0.373T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 + 0.595T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 0.154T + 59T^{2} \) |
| 61 | \( 1 - 2.86T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 + 6.55T + 79T^{2} \) |
| 83 | \( 1 - 0.649T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.83450207283160022493063895318, −6.97580516163783379459899992630, −6.25582484793828745653391230337, −5.61347633381562622065019089885, −5.08273023186997723416561901872, −4.13687462665261266753929763357, −3.65269322811716752399825608985, −2.51484763935340272779179294792, −1.76138245046110264342453076059, −0.32324561181329562083962375948,
0.32324561181329562083962375948, 1.76138245046110264342453076059, 2.51484763935340272779179294792, 3.65269322811716752399825608985, 4.13687462665261266753929763357, 5.08273023186997723416561901872, 5.61347633381562622065019089885, 6.25582484793828745653391230337, 6.97580516163783379459899992630, 7.83450207283160022493063895318
