| L(s) = 1 | + 2.47·3-s + 5-s + 0.527·7-s + 3.11·9-s + 3.11·11-s + 4.11·13-s + 2.47·15-s − 4.39·17-s − 3.70·19-s + 1.30·21-s − 23-s + 25-s + 0.284·27-s − 9.10·29-s + 4.83·31-s + 7.70·33-s + 0.527·35-s + 9.74·37-s + 10.1·39-s + 6.93·41-s − 4.45·43-s + 3.11·45-s + 0.642·47-s − 6.72·49-s − 10.8·51-s − 3.89·53-s + 3.11·55-s + ⋯ |
| L(s) = 1 | + 1.42·3-s + 0.447·5-s + 0.199·7-s + 1.03·9-s + 0.939·11-s + 1.14·13-s + 0.638·15-s − 1.06·17-s − 0.849·19-s + 0.284·21-s − 0.208·23-s + 0.200·25-s + 0.0546·27-s − 1.69·29-s + 0.867·31-s + 1.34·33-s + 0.0891·35-s + 1.60·37-s + 1.62·39-s + 1.08·41-s − 0.680·43-s + 0.464·45-s + 0.0936·47-s − 0.960·49-s − 1.52·51-s − 0.534·53-s + 0.420·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.889947735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.889947735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 7 | \( 1 - 0.527T + 7T^{2} \) |
| 11 | \( 1 - 3.11T + 11T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 - 9.74T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 - 0.642T + 47T^{2} \) |
| 53 | \( 1 + 3.89T + 53T^{2} \) |
| 59 | \( 1 - 8.79T + 59T^{2} \) |
| 61 | \( 1 + 3.45T + 61T^{2} \) |
| 67 | \( 1 + 8.60T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 5.81T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 5.43T + 89T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−9.731604345538169956288885548326, −9.143048666239080908928426854935, −8.538055073775153864000519605325, −7.81408982564360192177338553706, −6.67094484596823662163525989527, −5.95139527898640990195929300550, −4.39503789783715611643354618660, −3.71130921983602377177751038904, −2.52232716518289051615095279613, −1.57397861093359029755109714430,
1.57397861093359029755109714430, 2.52232716518289051615095279613, 3.71130921983602377177751038904, 4.39503789783715611643354618660, 5.95139527898640990195929300550, 6.67094484596823662163525989527, 7.81408982564360192177338553706, 8.538055073775153864000519605325, 9.143048666239080908928426854935, 9.731604345538169956288885548326
