L(s) = 1 | + 2-s + 1.85·3-s + 4-s + 0.0298·5-s + 1.85·6-s + 4.04·7-s + 8-s + 0.427·9-s + 0.0298·10-s + 0.274·11-s + 1.85·12-s − 5.56·13-s + 4.04·14-s + 0.0553·15-s + 16-s + 7.69·17-s + 0.427·18-s + 5.57·19-s + 0.0298·20-s + 7.49·21-s + 0.274·22-s − 6.49·23-s + 1.85·24-s − 4.99·25-s − 5.56·26-s − 4.76·27-s + 4.04·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.0133·5-s + 0.755·6-s + 1.52·7-s + 0.353·8-s + 0.142·9-s + 0.00945·10-s + 0.0826·11-s + 0.534·12-s − 1.54·13-s + 1.08·14-s + 0.0142·15-s + 0.250·16-s + 1.86·17-s + 0.100·18-s + 1.27·19-s + 0.00668·20-s + 1.63·21-s + 0.0584·22-s − 1.35·23-s + 0.377·24-s − 0.999·25-s − 1.09·26-s − 0.916·27-s + 0.764·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.306959484\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.306959484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 - 0.0298T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 - 0.274T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 - 7.69T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 1.55T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 9.08T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 3.99T + 67T^{2} \) |
| 71 | \( 1 - 0.743T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 - 16.0T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.029711889086269219923830519017, −7.68142283708153329651258567633, −7.51406745022898118774527953196, −5.89043663755233728836287000771, −5.46991076746644524207863653539, −4.55797132403059318571627168479, −3.89652821242534102067196525266, −2.85954744089074638668020343871, −2.28319884774996648856658471459, −1.25586528061182706924604965476,
1.25586528061182706924604965476, 2.28319884774996648856658471459, 2.85954744089074638668020343871, 3.89652821242534102067196525266, 4.55797132403059318571627168479, 5.46991076746644524207863653539, 5.89043663755233728836287000771, 7.51406745022898118774527953196, 7.68142283708153329651258567633, 8.029711889086269219923830519017