L(s) = 1 | + 2-s + 2.40·3-s + 4-s + 5-s + 2.40·6-s + 4.92·7-s + 8-s + 2.80·9-s + 10-s − 1.92·11-s + 2.40·12-s − 1.63·13-s + 4.92·14-s + 2.40·15-s + 16-s + 6.34·17-s + 2.80·18-s − 0.942·19-s + 20-s + 11.8·21-s − 1.92·22-s − 7.85·23-s + 2.40·24-s + 25-s − 1.63·26-s − 0.463·27-s + 4.92·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s + 0.447·5-s + 0.983·6-s + 1.86·7-s + 0.353·8-s + 0.935·9-s + 0.316·10-s − 0.580·11-s + 0.695·12-s − 0.454·13-s + 1.31·14-s + 0.622·15-s + 0.250·16-s + 1.53·17-s + 0.661·18-s − 0.216·19-s + 0.223·20-s + 2.58·21-s − 0.410·22-s − 1.63·23-s + 0.491·24-s + 0.200·25-s − 0.321·26-s − 0.0891·27-s + 0.930·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.501976449\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.501976449\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.40T + 3T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 1.63T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 19 | \( 1 + 0.942T + 19T^{2} \) |
| 23 | \( 1 + 7.85T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 + 0.366T + 31T^{2} \) |
| 37 | \( 1 - 8.44T + 37T^{2} \) |
| 41 | \( 1 - 5.10T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 + 0.0520T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 4.90T + 73T^{2} \) |
| 79 | \( 1 + 0.680T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.147891115439930617220323196606, −7.81266735432383532471866141433, −7.38744410547162740588018934014, −5.94966720907593133038559335616, −5.41311504192061575493132900307, −4.55961205362946424483073038919, −3.87262874733132414975391291305, −2.86355872409950217708127155076, −2.13384218808978587629153591602, −1.49109390791109276652340408851,
1.49109390791109276652340408851, 2.13384218808978587629153591602, 2.86355872409950217708127155076, 3.87262874733132414975391291305, 4.55961205362946424483073038919, 5.41311504192061575493132900307, 5.94966720907593133038559335616, 7.38744410547162740588018934014, 7.81266735432383532471866141433, 8.147891115439930617220323196606