| L(s) = 1 | − 5-s + 9-s + 11-s + 25-s + 2·31-s − 45-s − 49-s − 55-s − 2·59-s − 2·71-s + 81-s − 2·89-s + 99-s + ⋯ |
| L(s) = 1 | − 5-s + 9-s + 11-s + 25-s + 2·31-s − 45-s − 49-s − 55-s − 2·59-s − 2·71-s + 81-s − 2·89-s + 99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9532692914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9532692914\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.32815636617604335513665580214, −9.527704032426942790631538966813, −8.605454397704077497266261596498, −7.78208064002107078664097898908, −6.97309851376350357712805548297, −6.22631025778891259046329257916, −4.71019604333192610261163922125, −4.15001136503880216220993884264, −3.06934743235795448332471968416, −1.34130023847443283310441182436,
1.34130023847443283310441182436, 3.06934743235795448332471968416, 4.15001136503880216220993884264, 4.71019604333192610261163922125, 6.22631025778891259046329257916, 6.97309851376350357712805548297, 7.78208064002107078664097898908, 8.605454397704077497266261596498, 9.527704032426942790631538966813, 10.32815636617604335513665580214
