| L(s) = 1 | + 2.47·2-s + 0.453·3-s + 4.12·4-s + 1.41·5-s + 1.12·6-s + 1.07·7-s + 5.25·8-s − 2.79·9-s + 3.49·10-s + 11-s + 1.86·12-s − 5.36·13-s + 2.66·14-s + 0.639·15-s + 4.76·16-s + 7.50·17-s − 6.91·18-s − 2.87·19-s + 5.81·20-s + 0.487·21-s + 2.47·22-s − 0.298·23-s + 2.38·24-s − 3.00·25-s − 13.2·26-s − 2.62·27-s + 4.43·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 0.261·3-s + 2.06·4-s + 0.630·5-s + 0.457·6-s + 0.406·7-s + 1.85·8-s − 0.931·9-s + 1.10·10-s + 0.301·11-s + 0.539·12-s − 1.48·13-s + 0.711·14-s + 0.165·15-s + 1.19·16-s + 1.82·17-s − 1.63·18-s − 0.659·19-s + 1.30·20-s + 0.106·21-s + 0.527·22-s − 0.0622·23-s + 0.486·24-s − 0.601·25-s − 2.60·26-s − 0.505·27-s + 0.837·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.674394231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.674394231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 3 | \( 1 - 0.453T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 7.50T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 0.298T + 23T^{2} \) |
| 29 | \( 1 - 8.79T + 29T^{2} \) |
| 31 | \( 1 + 6.51T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 0.489T + 43T^{2} \) |
| 47 | \( 1 - 8.13T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 71 | \( 1 + 0.424T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−10.53008269591150535686066556943, −9.757554489373769839161502495402, −8.523769471553950503598873800733, −7.49063928886751103224490318797, −6.57036595155025577926987268448, −5.48617106691269999283867472643, −5.19637749167732064107102511868, −3.90809249016013474886063444286, −2.90619760005937041054484091450, −2.00105888022366345459772104945,
2.00105888022366345459772104945, 2.90619760005937041054484091450, 3.90809249016013474886063444286, 5.19637749167732064107102511868, 5.48617106691269999283867472643, 6.57036595155025577926987268448, 7.49063928886751103224490318797, 8.523769471553950503598873800733, 9.757554489373769839161502495402, 10.53008269591150535686066556943
