| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 2·5-s − 6·6-s − 10·7-s − 8·8-s + 9·9-s − 4·10-s − 10·11-s + 12·12-s + 20·14-s + 6·15-s + 16·16-s − 18·18-s + 8·20-s − 30·21-s + 20·22-s − 24·24-s − 21·25-s + 27·27-s − 40·28-s + 50·29-s − 12·30-s + 38·31-s − 32·32-s − 30·33-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 2/5·5-s − 6-s − 1.42·7-s − 8-s + 9-s − 2/5·10-s − 0.909·11-s + 12-s + 10/7·14-s + 2/5·15-s + 16-s − 18-s + 2/5·20-s − 1.42·21-s + 0.909·22-s − 24-s − 0.839·25-s + 27-s − 1.42·28-s + 1.72·29-s − 2/5·30-s + 1.22·31-s − 32-s − 0.909·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7748540824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7748540824\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 T + p^{2} T^{2} \) |
| 7 | \( 1 + 10 T + p^{2} T^{2} \) |
| 11 | \( 1 + 10 T + p^{2} T^{2} \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 50 T + p^{2} T^{2} \) |
| 31 | \( 1 - 38 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 94 T + p^{2} T^{2} \) |
| 59 | \( 1 + 10 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 50 T + p^{2} T^{2} \) |
| 79 | \( 1 + 58 T + p^{2} T^{2} \) |
| 83 | \( 1 - 134 T + p^{2} T^{2} \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 190 T + p^{2} T^{2} \) |
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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
−17.71836482170190417794677250227, −16.14254629840978342810362187323, −15.46297586969984238831661207897, −13.72056564015015455118232174861, −12.48559668706853115991333872450, −10.28767870342943400895606247536, −9.508946168495398961435325029259, −8.091366243197074677376278444522, −6.53073453352690696705865240848, −2.84975734802491852002940479569,
2.84975734802491852002940479569, 6.53073453352690696705865240848, 8.091366243197074677376278444522, 9.508946168495398961435325029259, 10.28767870342943400895606247536, 12.48559668706853115991333872450, 13.72056564015015455118232174861, 15.46297586969984238831661207897, 16.14254629840978342810362187323, 17.71836482170190417794677250227
