L(s) = 1 | − 1.67·2-s − 1.98·3-s + 1.79·4-s + 3.31·6-s + 1.19·7-s − 1.32·8-s + 2.94·9-s − 3.56·12-s − 1.99·14-s + 0.419·16-s − 4.92·18-s − 0.116·19-s − 2.37·21-s − 0.573·23-s + 2.62·24-s + 25-s − 3.86·27-s + 2.14·28-s + 0.347·31-s + 0.622·32-s + 5.27·36-s − 1.87·37-s + 0.194·38-s + 1.53·41-s + 3.96·42-s + 1.78·43-s + 0.958·46-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 1.98·3-s + 1.79·4-s + 3.31·6-s + 1.19·7-s − 1.32·8-s + 2.94·9-s − 3.56·12-s − 1.99·14-s + 0.419·16-s − 4.92·18-s − 0.116·19-s − 2.37·21-s − 0.573·23-s + 2.62·24-s + 25-s − 3.86·27-s + 2.14·28-s + 0.347·31-s + 0.622·32-s + 5.27·36-s − 1.87·37-s + 0.194·38-s + 1.53·41-s + 3.96·42-s + 1.78·43-s + 0.958·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2827976740\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2827976740\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 - T \) |
good | 2 | \( 1 + 1.67T + T^{2} \) |
| 3 | \( 1 + 1.98T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.19T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.116T + T^{2} \) |
| 23 | \( 1 + 0.573T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.347T + T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 - 1.53T + T^{2} \) |
| 43 | \( 1 - 1.78T + T^{2} \) |
| 47 | \( 1 + 1.37T + T^{2} \) |
| 53 | \( 1 - 1.78T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.792T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.42379762146575837487770626567, −9.565955077626590888003939068111, −8.558546041458691680546928728267, −7.61530957414097435706331489665, −7.00528834362728803323949372960, −6.12556144929196497167190105955, −5.19101419326351148503661645241, −4.32495861042507910678864118183, −1.92939925858628062822630998349, −0.914489418029993841565598819664,
0.914489418029993841565598819664, 1.92939925858628062822630998349, 4.32495861042507910678864118183, 5.19101419326351148503661645241, 6.12556144929196497167190105955, 7.00528834362728803323949372960, 7.61530957414097435706331489665, 8.558546041458691680546928728267, 9.565955077626590888003939068111, 10.42379762146575837487770626567