L(s) = 1 | − 2.37·2-s − 3.14·3-s + 3.64·4-s + 0.257·5-s + 7.48·6-s + 1.36·7-s − 3.90·8-s + 6.91·9-s − 0.611·10-s − 3.32·11-s − 11.4·12-s − 0.510·13-s − 3.23·14-s − 0.810·15-s + 1.98·16-s − 5.42·17-s − 16.4·18-s − 5.24·19-s + 0.938·20-s − 4.28·21-s + 7.90·22-s + 1.00·23-s + 12.2·24-s − 4.93·25-s + 1.21·26-s − 12.3·27-s + 4.96·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 1.81·3-s + 1.82·4-s + 0.115·5-s + 3.05·6-s + 0.514·7-s − 1.37·8-s + 2.30·9-s − 0.193·10-s − 1.00·11-s − 3.31·12-s − 0.141·13-s − 0.864·14-s − 0.209·15-s + 0.496·16-s − 1.31·17-s − 3.87·18-s − 1.20·19-s + 0.209·20-s − 0.936·21-s + 1.68·22-s + 0.208·23-s + 2.50·24-s − 0.986·25-s + 0.237·26-s − 2.37·27-s + 0.937·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06387245058\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06387245058\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4027 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 0.257T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 3.32T + 11T^{2} \) |
| 13 | \( 1 + 0.510T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 19 | \( 1 + 5.24T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 29 | \( 1 + 1.81T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 + 7.60T + 41T^{2} \) |
| 43 | \( 1 + 5.13T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 8.61T + 61T^{2} \) |
| 67 | \( 1 + 5.63T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 5.01T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−8.508502312794903181847381223889, −7.65012217559226135791784672689, −7.05771872616748512771837638242, −6.46147946437348788361799381826, −5.63471115803788385784884677269, −4.96646465528633399501909433322, −4.06790340196555556513909505481, −2.23039780444699698599129598353, −1.61443933412807182200637316357, −0.20275899913021850455650201835,
0.20275899913021850455650201835, 1.61443933412807182200637316357, 2.23039780444699698599129598353, 4.06790340196555556513909505481, 4.96646465528633399501909433322, 5.63471115803788385784884677269, 6.46147946437348788361799381826, 7.05771872616748512771837638242, 7.65012217559226135791784672689, 8.508502312794903181847381223889