L(s) = 1 | − 0.116·2-s + 1.19·3-s − 0.986·4-s − 0.138·6-s + 0.792·7-s + 0.231·8-s + 0.426·9-s − 1.17·12-s − 0.0921·14-s + 0.959·16-s − 0.0495·18-s + 1.78·19-s + 0.946·21-s − 1.37·23-s + 0.275·24-s + 25-s − 0.685·27-s − 0.781·28-s + 0.347·31-s − 0.342·32-s − 0.420·36-s − 1.87·37-s − 0.207·38-s + 1.53·41-s − 0.110·42-s − 1.67·43-s + 0.159·46-s + ⋯ |
L(s) = 1 | − 0.116·2-s + 1.19·3-s − 0.986·4-s − 0.138·6-s + 0.792·7-s + 0.231·8-s + 0.426·9-s − 1.17·12-s − 0.0921·14-s + 0.959·16-s − 0.0495·18-s + 1.78·19-s + 0.946·21-s − 1.37·23-s + 0.275·24-s + 25-s − 0.685·27-s − 0.781·28-s + 0.347·31-s − 0.342·32-s − 0.420·36-s − 1.87·37-s − 0.207·38-s + 1.53·41-s − 0.110·42-s − 1.67·43-s + 0.159·46-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.227170521\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.227170521\) |
\(L(1)\) |
|
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 | \( 1 + 0.116T + T^{2} \) |
| 3 | \( 1 - 1.19T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 0.792T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.78T + T^{2} \) |
| 23 | \( 1 + 1.37T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 0.347T + T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 - 1.53T + T^{2} \) |
| 43 | \( 1 + 1.67T + T^{2} \) |
| 47 | \( 1 - 1.94T + T^{2} \) |
| 53 | \( 1 + 1.67T + T^{2} \) |
| 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 | \( 1 + 1.98T + T^{2} \) |
| 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
−9.843493989929148960598492976864, −9.311648883950822854084801764072, −8.454170294418607788866434192844, −8.027832134127019535465875535663, −7.21361135051369251398909862127, −5.67609280776698598832633927088, −4.85015688062753448610314471449, −3.84405139581806485856280208519, −2.95587660680363076836170497980, −1.53106581503334650470559092247,
1.53106581503334650470559092247, 2.95587660680363076836170497980, 3.84405139581806485856280208519, 4.85015688062753448610314471449, 5.67609280776698598832633927088, 7.21361135051369251398909862127, 8.027832134127019535465875535663, 8.454170294418607788866434192844, 9.311648883950822854084801764072, 9.843493989929148960598492976864