| L(s) = 1 | − 1.40·3-s + 1.41·5-s − 3.63·7-s − 1.03·9-s + 1.41·11-s + 4.78·13-s − 1.98·15-s + 4.74·17-s + 0.219·19-s + 5.08·21-s − 8.30·23-s − 2.99·25-s + 5.65·27-s − 8.51·29-s + 4.94·31-s − 1.98·33-s − 5.14·35-s − 8.81·37-s − 6.70·39-s + 7.32·41-s − 1.68·43-s − 1.46·45-s + 10.3·47-s + 6.18·49-s − 6.64·51-s + 5.38·53-s + 2.00·55-s + ⋯ |
| L(s) = 1 | − 0.809·3-s + 0.633·5-s − 1.37·7-s − 0.345·9-s + 0.426·11-s + 1.32·13-s − 0.512·15-s + 1.15·17-s + 0.0504·19-s + 1.11·21-s − 1.73·23-s − 0.598·25-s + 1.08·27-s − 1.58·29-s + 0.888·31-s − 0.345·33-s − 0.869·35-s − 1.44·37-s − 1.07·39-s + 1.14·41-s − 0.256·43-s − 0.218·45-s + 1.51·47-s + 0.883·49-s − 0.930·51-s + 0.740·53-s + 0.270·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143905653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143905653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 3 | \( 1 + 1.40T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 3.63T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 4.78T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 0.219T + 19T^{2} \) |
| 23 | \( 1 + 8.30T + 23T^{2} \) |
| 29 | \( 1 + 8.51T + 29T^{2} \) |
| 31 | \( 1 - 4.94T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 + 1.68T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.38T + 53T^{2} \) |
| 59 | \( 1 + 9.25T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 - 6.01T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.04T + 73T^{2} \) |
| 79 | \( 1 - 8.82T + 79T^{2} \) |
| 83 | \( 1 - 0.697T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.71473109213233593786445308644, −6.77474194078752975617988857881, −6.22576081031133708383893491451, −5.73811359146776529754370105993, −5.46514455906616137339100976096, −3.99185134910981690743342969465, −3.64715235109144675105465179335, −2.67949942311349776400944358205, −1.61913563641591029113604668950, −0.54708575173863111238632238605,
0.54708575173863111238632238605, 1.61913563641591029113604668950, 2.67949942311349776400944358205, 3.64715235109144675105465179335, 3.99185134910981690743342969465, 5.46514455906616137339100976096, 5.73811359146776529754370105993, 6.22576081031133708383893491451, 6.77474194078752975617988857881, 7.71473109213233593786445308644
