| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s − 2·13-s + 15-s + 2·17-s + 19-s + 4·23-s + 25-s + 27-s − 6·29-s + 4·31-s − 4·33-s + 6·37-s − 2·39-s + 10·41-s + 4·43-s + 45-s − 12·47-s − 7·49-s + 2·51-s − 6·53-s − 4·55-s + 57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.132·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.705277752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.705277752\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.89464989295771, −15.07816321685752, −14.59442088986342, −14.37033066069411, −13.32316980120882, −13.22489445823221, −12.68394389881669, −12.01402825732025, −11.14554627873736, −10.85763009564661, −9.910435922832978, −9.735616207679410, −9.118133164230526, −8.317987874729830, −7.779942748725993, −7.401982058930073, −6.577293129709557, −5.882188157079488, −5.169727201994963, −4.731931019718075, −3.807086926759255, −2.955141008483067, −2.552641598593329, −1.704472432998404, −0.6616482933578650,
0.6616482933578650, 1.704472432998404, 2.552641598593329, 2.955141008483067, 3.807086926759255, 4.731931019718075, 5.169727201994963, 5.882188157079488, 6.577293129709557, 7.401982058930073, 7.779942748725993, 8.317987874729830, 9.118133164230526, 9.735616207679410, 9.910435922832978, 10.85763009564661, 11.14554627873736, 12.01402825732025, 12.68394389881669, 13.22489445823221, 13.32316980120882, 14.37033066069411, 14.59442088986342, 15.07816321685752, 15.89464989295771
