| L(s) = 1 | − 3·3-s + 5·5-s + 9·9-s + 4·11-s + 54·13-s − 15·15-s + 114·17-s + 44·19-s + 96·23-s + 25·25-s − 27·27-s + 134·29-s − 272·31-s − 12·33-s − 98·37-s − 162·39-s − 6·41-s + 12·43-s + 45·45-s − 200·47-s − 343·49-s − 342·51-s + 654·53-s + 20·55-s − 132·57-s + 36·59-s − 442·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.109·11-s + 1.15·13-s − 0.258·15-s + 1.62·17-s + 0.531·19-s + 0.870·23-s + 1/5·25-s − 0.192·27-s + 0.858·29-s − 1.57·31-s − 0.0633·33-s − 0.435·37-s − 0.665·39-s − 0.0228·41-s + 0.0425·43-s + 0.149·45-s − 0.620·47-s − 49-s − 0.939·51-s + 1.69·53-s + 0.0490·55-s − 0.306·57-s + 0.0794·59-s − 0.927·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.548962721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.548962721\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 272 T + p^{3} T^{2} \) |
| 37 | \( 1 + 98 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 200 T + p^{3} T^{2} \) |
| 53 | \( 1 - 654 T + p^{3} T^{2} \) |
| 59 | \( 1 - 36 T + p^{3} T^{2} \) |
| 61 | \( 1 + 442 T + p^{3} T^{2} \) |
| 67 | \( 1 + 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 632 T + p^{3} T^{2} \) |
| 73 | \( 1 + 390 T + p^{3} T^{2} \) |
| 79 | \( 1 - 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 694 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
| 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
−12.95838307436294775846390531527, −11.95050364518667877142230158441, −10.91722180508417800488815687848, −9.974507179172154889466206720584, −8.810996478859550380947159512961, −7.42113699043004824488959684736, −6.14172766315947894943965130321, −5.17573408770574063463958732887, −3.43539994121616161817867703262, −1.24569809277041399342347338707,
1.24569809277041399342347338707, 3.43539994121616161817867703262, 5.17573408770574063463958732887, 6.14172766315947894943965130321, 7.42113699043004824488959684736, 8.810996478859550380947159512961, 9.974507179172154889466206720584, 10.91722180508417800488815687848, 11.95050364518667877142230158441, 12.95838307436294775846390531527
