| L(s) = 1 | + 1.36·3-s + 4.14·5-s − 1.14·9-s − 5.64·11-s − 2.86·13-s + 5.64·15-s + 2·17-s + 4.28·19-s − 2.72·23-s + 12.1·25-s − 5.64·27-s + 29-s − 5.36·31-s − 7.69·33-s − 6.28·37-s − 3.91·39-s + 11.7·41-s + 2.91·43-s − 4.72·45-s − 4.19·47-s − 7·49-s + 2.72·51-s + 1.41·53-s − 23.3·55-s + 5.83·57-s + 1.27·59-s + 3.45·61-s + ⋯ |
| L(s) = 1 | + 0.787·3-s + 1.85·5-s − 0.380·9-s − 1.70·11-s − 0.795·13-s + 1.45·15-s + 0.485·17-s + 0.982·19-s − 0.568·23-s + 2.43·25-s − 1.08·27-s + 0.185·29-s − 0.963·31-s − 1.33·33-s − 1.03·37-s − 0.626·39-s + 1.83·41-s + 0.445·43-s − 0.704·45-s − 0.611·47-s − 49-s + 0.381·51-s + 0.194·53-s − 3.15·55-s + 0.773·57-s + 0.165·59-s + 0.442·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780930612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780930612\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 2.86T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 + 6.28T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 - 2.91T + 43T^{2} \) |
| 47 | \( 1 + 4.19T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 - 3.45T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 7.73T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.27T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−12.56052927211091795154011581597, −10.98117942127896842888941868706, −9.948372895884767339753466196864, −9.504469075012756090621857268588, −8.330997280798521711529752017256, −7.31330664726278357779038422986, −5.76646473459184482627566521212, −5.20135792932851369760585826873, −2.97660797935870030227489911056, −2.14299931080491498205771118542,
2.14299931080491498205771118542, 2.97660797935870030227489911056, 5.20135792932851369760585826873, 5.76646473459184482627566521212, 7.31330664726278357779038422986, 8.330997280798521711529752017256, 9.504469075012756090621857268588, 9.948372895884767339753466196864, 10.98117942127896842888941868706, 12.56052927211091795154011581597
