| L(s) = 1 | + 1.81·3-s − 2.70·5-s + 3.36·7-s + 0.298·9-s + 5.17·11-s − 13-s − 4.90·15-s + 6.70·17-s − 5.17·19-s + 6.10·21-s + 2.29·25-s − 4.90·27-s + 2·29-s + 8.80·31-s + 9.40·33-s − 9.08·35-s − 2.70·37-s − 1.81·39-s + 3.40·41-s + 8.53·43-s − 0.806·45-s + 3.36·47-s + 4.29·49-s + 12.1·51-s + 11.4·53-s − 13.9·55-s − 9.40·57-s + ⋯ |
| L(s) = 1 | + 1.04·3-s − 1.20·5-s + 1.27·7-s + 0.0994·9-s + 1.56·11-s − 0.277·13-s − 1.26·15-s + 1.62·17-s − 1.18·19-s + 1.33·21-s + 0.459·25-s − 0.944·27-s + 0.371·29-s + 1.58·31-s + 1.63·33-s − 1.53·35-s − 0.444·37-s − 0.290·39-s + 0.531·41-s + 1.30·43-s − 0.120·45-s + 0.490·47-s + 0.614·49-s + 1.70·51-s + 1.56·53-s − 1.88·55-s − 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.162494045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.162494045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 5.17T + 11T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 5.17T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 3.40T + 41T^{2} \) |
| 43 | \( 1 - 8.53T + 43T^{2} \) |
| 47 | \( 1 - 3.36T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 2.08T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−10.16523451672448072205636302862, −9.053062251090870886091780690810, −8.415205330558826404295549873142, −7.88523734618206710576870221831, −7.13235245164607957783453938055, −5.81751617008584883977496739235, −4.39832950384529285249577259859, −3.91951633772264242810014112558, −2.74143613165775722661326551162, −1.30426678231658821999703341814,
1.30426678231658821999703341814, 2.74143613165775722661326551162, 3.91951633772264242810014112558, 4.39832950384529285249577259859, 5.81751617008584883977496739235, 7.13235245164607957783453938055, 7.88523734618206710576870221831, 8.415205330558826404295549873142, 9.053062251090870886091780690810, 10.16523451672448072205636302862
