| L(s) = 1 | − 2.30·3-s − 0.697·7-s + 2.30·9-s + 11-s + 5·13-s + 6.90·17-s + 19-s + 1.60·21-s + 7.30·23-s + 1.60·27-s + 0.908·29-s − 10.2·31-s − 2.30·33-s + 2.39·37-s − 11.5·39-s − 5.60·41-s − 7.21·43-s + 3·47-s − 6.51·49-s − 15.9·51-s − 1.30·53-s − 2.30·57-s + 14.2·59-s − 7.90·61-s − 1.60·63-s + 4·67-s − 16.8·69-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 0.263·7-s + 0.767·9-s + 0.301·11-s + 1.38·13-s + 1.67·17-s + 0.229·19-s + 0.350·21-s + 1.52·23-s + 0.308·27-s + 0.168·29-s − 1.83·31-s − 0.400·33-s + 0.393·37-s − 1.84·39-s − 0.875·41-s − 1.09·43-s + 0.437·47-s − 0.930·49-s − 2.22·51-s − 0.178·53-s − 0.305·57-s + 1.85·59-s − 1.01·61-s − 0.202·63-s + 0.488·67-s − 2.02·69-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
| \(L(1)\) |
\(\approx\) |
\(1.284296843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.284296843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 + 0.697T + 7T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 - 0.908T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 1.30T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 2.60T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.51T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.383710577061519445185920723820, −7.45113519041401167425207392654, −6.73519063502071224646238258724, −6.10934356303004812196802865034, −5.43770016815597989292616937161, −4.95148773893296889525470139011, −3.71570832216853711631870881021, −3.19310364069498926035373923900, −1.53790504451770034547893431848, −0.74294580126750192374148823646,
0.74294580126750192374148823646, 1.53790504451770034547893431848, 3.19310364069498926035373923900, 3.71570832216853711631870881021, 4.95148773893296889525470139011, 5.43770016815597989292616937161, 6.10934356303004812196802865034, 6.73519063502071224646238258724, 7.45113519041401167425207392654, 8.383710577061519445185920723820
