| L(s) = 1 | + 0.381·3-s + 0.331·5-s − 2.55·7-s − 2.85·9-s − 3.59·11-s − 13-s + 0.126·15-s − 2.36·17-s + 6.73·19-s − 0.976·21-s + 4.79·23-s − 4.88·25-s − 2.23·27-s + 29-s − 3.61·31-s − 1.37·33-s − 0.849·35-s − 5.90·37-s − 0.381·39-s + 3.31·41-s + 3.25·43-s − 0.947·45-s + 8.01·47-s − 0.446·49-s − 0.903·51-s + 0.428·53-s − 1.19·55-s + ⋯ |
| L(s) = 1 | + 0.220·3-s + 0.148·5-s − 0.967·7-s − 0.951·9-s − 1.08·11-s − 0.277·13-s + 0.0326·15-s − 0.574·17-s + 1.54·19-s − 0.213·21-s + 1.00·23-s − 0.977·25-s − 0.429·27-s + 0.185·29-s − 0.649·31-s − 0.238·33-s − 0.143·35-s − 0.970·37-s − 0.0610·39-s + 0.517·41-s + 0.496·43-s − 0.141·45-s + 1.16·47-s − 0.0638·49-s − 0.126·51-s + 0.0588·53-s − 0.161·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132803535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132803535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 0.331T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 17 | \( 1 + 2.36T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 + 5.90T + 37T^{2} \) |
| 41 | \( 1 - 3.31T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 - 0.428T + 53T^{2} \) |
| 59 | \( 1 - 3.20T + 59T^{2} \) |
| 61 | \( 1 + 9.83T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 2.02T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.996472170046395855955182656508, −7.40328061975577369852551554323, −6.74720044756460054373299632397, −5.67377153577026436313973180106, −5.51453778400573475427842247685, −4.46317443010098851101239503323, −3.30521427849081277403695340272, −2.95428313124823018442784709729, −2.05410441343237411566038885066, −0.52030966943061468586032957157,
0.52030966943061468586032957157, 2.05410441343237411566038885066, 2.95428313124823018442784709729, 3.30521427849081277403695340272, 4.46317443010098851101239503323, 5.51453778400573475427842247685, 5.67377153577026436313973180106, 6.74720044756460054373299632397, 7.40328061975577369852551554323, 7.996472170046395855955182656508
