| L(s) = 1 | − 3-s + 4·7-s + 9-s − 13-s − 3·19-s − 4·21-s − 4·23-s − 27-s − 29-s − 8·31-s − 37-s + 39-s + 41-s + 6·43-s − 11·47-s + 9·49-s − 3·53-s + 3·57-s + 10·59-s + 4·61-s + 4·63-s + 13·67-s + 4·69-s + 9·71-s − 3·79-s + 81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.688·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 0.185·29-s − 1.43·31-s − 0.164·37-s + 0.160·39-s + 0.156·41-s + 0.914·43-s − 1.60·47-s + 9/7·49-s − 0.412·53-s + 0.397·57-s + 1.30·59-s + 0.512·61-s + 0.503·63-s + 1.58·67-s + 0.481·69-s + 1.06·71-s − 0.337·79-s + 1/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.25001653082740, −14.78009710490683, −14.34456467569658, −13.95662663142767, −13.09047993988224, −12.67738578576728, −12.14721280612154, −11.37701372316011, −11.28577838312373, −10.68380774044485, −10.07815054676663, −9.481763675566326, −8.795229019175465, −8.134446967493183, −7.838061023586729, −7.092756445056883, −6.557918088860335, −5.751876579182332, −5.288547573593663, −4.755173009300293, −4.117688667997870, −3.534190641220321, −2.281658343149351, −1.918141075741189, −1.048625806007888, 0,
1.048625806007888, 1.918141075741189, 2.281658343149351, 3.534190641220321, 4.117688667997870, 4.755173009300293, 5.288547573593663, 5.751876579182332, 6.557918088860335, 7.092756445056883, 7.838061023586729, 8.134446967493183, 8.795229019175465, 9.481763675566326, 10.07815054676663, 10.68380774044485, 11.28577838312373, 11.37701372316011, 12.14721280612154, 12.67738578576728, 13.09047993988224, 13.95662663142767, 14.34456467569658, 14.78009710490683, 15.25001653082740
