L(s) = 1 | − 2-s − 1.61·3-s + 4-s + 0.618·5-s + 1.61·6-s − 7-s − 8-s − 0.381·9-s − 0.618·10-s + 11-s − 1.61·12-s − 13-s + 14-s − 1.00·15-s + 16-s − 0.854·17-s + 0.381·18-s + 2.61·19-s + 0.618·20-s + 1.61·21-s − 22-s − 3.23·23-s + 1.61·24-s − 4.61·25-s + 26-s + 5.47·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.934·3-s + 0.5·4-s + 0.276·5-s + 0.660·6-s − 0.377·7-s − 0.353·8-s − 0.127·9-s − 0.195·10-s + 0.301·11-s − 0.467·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s − 0.207·17-s + 0.0900·18-s + 0.600·19-s + 0.138·20-s + 0.353·21-s − 0.213·22-s − 0.674·23-s + 0.330·24-s − 0.923·25-s + 0.196·26-s + 1.05·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6669943873\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6669943873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 3.32T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 + 8.79T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 0.854T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−9.235963682818133979728299624095, −8.484425194986561866567330003381, −7.53248313322933893553171838680, −6.78234583415087013994058125925, −5.98178546782377360012994429630, −5.48576170842411076389273389352, −4.32699356708660487510163236422, −3.13479813509626073350065211238, −1.99090890199190097518095670381, −0.61878896339752247411944796554,
0.61878896339752247411944796554, 1.99090890199190097518095670381, 3.13479813509626073350065211238, 4.32699356708660487510163236422, 5.48576170842411076389273389352, 5.98178546782377360012994429630, 6.78234583415087013994058125925, 7.53248313322933893553171838680, 8.484425194986561866567330003381, 9.235963682818133979728299624095