L(s) = 1 | − 2-s − 1.09·3-s + 4-s − 3.08·5-s + 1.09·6-s + 3.13·7-s − 8-s − 1.79·9-s + 3.08·10-s − 2.44·11-s − 1.09·12-s − 0.162·13-s − 3.13·14-s + 3.39·15-s + 16-s − 17-s + 1.79·18-s + 3.47·19-s − 3.08·20-s − 3.45·21-s + 2.44·22-s − 5.13·23-s + 1.09·24-s + 4.50·25-s + 0.162·26-s + 5.26·27-s + 3.13·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.635·3-s + 0.5·4-s − 1.37·5-s + 0.449·6-s + 1.18·7-s − 0.353·8-s − 0.596·9-s + 0.975·10-s − 0.738·11-s − 0.317·12-s − 0.0451·13-s − 0.838·14-s + 0.875·15-s + 0.250·16-s − 0.242·17-s + 0.421·18-s + 0.798·19-s − 0.689·20-s − 0.753·21-s + 0.521·22-s − 1.06·23-s + 0.224·24-s + 0.901·25-s + 0.0319·26-s + 1.01·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4982925947\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4982925947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 1.09T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 - 3.13T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.162T + 13T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 23 | \( 1 + 5.13T + 23T^{2} \) |
| 29 | \( 1 + 6.96T + 29T^{2} \) |
| 31 | \( 1 + 0.948T + 31T^{2} \) |
| 37 | \( 1 + 0.764T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 61 | \( 1 - 2.93T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 2.43T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 5.99T + 89T^{2} \) |
| 97 | \( 1 + 7.05T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.901528951193328926565669343551, −8.276908315905364978389905517781, −7.68197011027314153026930004882, −7.18904807462609116264775695788, −5.90547341252748112255683398885, −5.21986356775422155699433362339, −4.31008956469139151970492942210, −3.26733445303268290896807481638, −2.00026879934285272182343840487, −0.52318597111152583342267472842,
0.52318597111152583342267472842, 2.00026879934285272182343840487, 3.26733445303268290896807481638, 4.31008956469139151970492942210, 5.21986356775422155699433362339, 5.90547341252748112255683398885, 7.18904807462609116264775695788, 7.68197011027314153026930004882, 8.276908315905364978389905517781, 8.901528951193328926565669343551