| L(s) = 1 | − 2.59·3-s + 2.72·7-s + 3.72·9-s − 11-s − 13-s + 4.59·17-s − 8.18·19-s − 7.05·21-s + 0.407·23-s − 1.87·27-s + 7.46·29-s + 4.44·31-s + 2.59·33-s + 7.31·37-s + 2.59·39-s − 3.31·41-s − 7.49·43-s + 7.05·47-s + 0.407·49-s − 11.9·51-s − 0.979·53-s + 21.2·57-s − 7.05·59-s − 4.46·61-s + 10.1·63-s − 2.25·67-s − 1.05·69-s + ⋯ |
| L(s) = 1 | − 1.49·3-s + 1.02·7-s + 1.24·9-s − 0.301·11-s − 0.277·13-s + 1.11·17-s − 1.87·19-s − 1.53·21-s + 0.0849·23-s − 0.360·27-s + 1.38·29-s + 0.798·31-s + 0.451·33-s + 1.20·37-s + 0.415·39-s − 0.517·41-s − 1.14·43-s + 1.02·47-s + 0.0581·49-s − 1.66·51-s − 0.134·53-s + 2.81·57-s − 0.918·59-s − 0.571·61-s + 1.27·63-s − 0.275·67-s − 0.127·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092696478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092696478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.59T + 3T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 8.18T + 19T^{2} \) |
| 23 | \( 1 - 0.407T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 - 7.05T + 47T^{2} \) |
| 53 | \( 1 + 0.979T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 + 4.46T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 3.14T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−8.153689768964519943085900558485, −7.71418429749629375600534522640, −6.58484054521497817913423600596, −6.24203055441219086352119797438, −5.31774155116697038803070636477, −4.78659835844888236540871457665, −4.21010478820667829163358344450, −2.82479873049171890971264238986, −1.67995520174280153959856151311, −0.65452782808245726837733045522,
0.65452782808245726837733045522, 1.67995520174280153959856151311, 2.82479873049171890971264238986, 4.21010478820667829163358344450, 4.78659835844888236540871457665, 5.31774155116697038803070636477, 6.24203055441219086352119797438, 6.58484054521497817913423600596, 7.71418429749629375600534522640, 8.153689768964519943085900558485
