| L(s) = 1 | − 1.19·3-s − 0.449·5-s − 2.62·7-s − 1.57·9-s − 0.634·11-s + 0.888·13-s + 0.536·15-s − 4.11·17-s − 7.30·19-s + 3.12·21-s − 3.95·23-s − 4.79·25-s + 5.45·27-s − 9.40·29-s + 10.3·31-s + 0.756·33-s + 1.18·35-s − 3.75·37-s − 1.05·39-s − 10.1·41-s − 6.79·43-s + 0.710·45-s + 4.84·47-s − 0.110·49-s + 4.89·51-s − 7.23·53-s + 0.285·55-s + ⋯ |
| L(s) = 1 | − 0.688·3-s − 0.201·5-s − 0.992·7-s − 0.526·9-s − 0.191·11-s + 0.246·13-s + 0.138·15-s − 0.996·17-s − 1.67·19-s + 0.682·21-s − 0.824·23-s − 0.959·25-s + 1.05·27-s − 1.74·29-s + 1.85·31-s + 0.131·33-s + 0.199·35-s − 0.617·37-s − 0.169·39-s − 1.59·41-s − 1.03·43-s + 0.105·45-s + 0.706·47-s − 0.0157·49-s + 0.686·51-s − 0.993·53-s + 0.0385·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07893839097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07893839097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 + 0.449T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 + 0.634T + 11T^{2} \) |
| 13 | \( 1 - 0.888T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 + 7.30T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.79T + 43T^{2} \) |
| 47 | \( 1 - 4.84T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 - 3.95T + 71T^{2} \) |
| 73 | \( 1 + 8.81T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 8.50T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−7.919386077331815728360545005435, −6.81154680000630862576234062704, −6.44879490545706676234800082847, −5.90840432144188388741332327943, −5.12188780064479604759027004703, −4.24705212448273000190872612773, −3.61733535555531650628752366017, −2.65982738185421841962403721427, −1.81489162260801278993000989726, −0.13474294985557140333572466622,
0.13474294985557140333572466622, 1.81489162260801278993000989726, 2.65982738185421841962403721427, 3.61733535555531650628752366017, 4.24705212448273000190872612773, 5.12188780064479604759027004703, 5.90840432144188388741332327943, 6.44879490545706676234800082847, 6.81154680000630862576234062704, 7.919386077331815728360545005435
