| L(s) = 1 | − 2·5-s + 7-s − 3·13-s + 2·17-s + 4·19-s + 7·23-s − 25-s − 3·29-s + 3·31-s − 2·35-s + 3·41-s + 43-s − 6·47-s + 49-s − 2·53-s − 3·59-s + 5·61-s + 6·65-s − 13·67-s + 9·71-s − 16·79-s + 11·83-s − 4·85-s − 89-s − 3·91-s − 8·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 0.832·13-s + 0.485·17-s + 0.917·19-s + 1.45·23-s − 1/5·25-s − 0.557·29-s + 0.538·31-s − 0.338·35-s + 0.468·41-s + 0.152·43-s − 0.875·47-s + 1/7·49-s − 0.274·53-s − 0.390·59-s + 0.640·61-s + 0.744·65-s − 1.58·67-s + 1.06·71-s − 1.80·79-s + 1.20·83-s − 0.433·85-s − 0.105·89-s − 0.314·91-s − 0.820·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.82192257487659, −13.23891913146184, −12.80957783419219, −12.24786970000810, −11.80717713306570, −11.46859127290659, −11.01103913766845, −10.44084104050854, −9.848942502394104, −9.429905607277845, −8.891695738840146, −8.315463207466776, −7.727430987882075, −7.476391635419486, −7.030598643353374, −6.336701509684591, −5.661397692502545, −5.095467747536173, −4.717559859354605, −4.111765919594613, −3.426586554769597, −3.012058826300714, −2.330642929668391, −1.472024769740552, −0.8379396503838956, 0,
0.8379396503838956, 1.472024769740552, 2.330642929668391, 3.012058826300714, 3.426586554769597, 4.111765919594613, 4.717559859354605, 5.095467747536173, 5.661397692502545, 6.336701509684591, 7.030598643353374, 7.476391635419486, 7.727430987882075, 8.315463207466776, 8.891695738840146, 9.429905607277845, 9.848942502394104, 10.44084104050854, 11.01103913766845, 11.46859127290659, 11.80717713306570, 12.24786970000810, 12.80957783419219, 13.23891913146184, 13.82192257487659
