| L(s) = 1 | − 2·2-s + 4·4-s + 15.8·5-s − 8·8-s − 31.6·10-s − 51.8·11-s + 38.8·13-s + 16·16-s − 27.3·17-s + 76.5·19-s + 63.3·20-s + 103.·22-s − 147.·23-s + 125.·25-s − 77.6·26-s − 240.·29-s − 296.·31-s − 32·32-s + 54.6·34-s − 161.·37-s − 153.·38-s − 126.·40-s + 102.·41-s − 328.·43-s − 207.·44-s + 294.·46-s − 67.9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.41·5-s − 0.353·8-s − 1.00·10-s − 1.42·11-s + 0.828·13-s + 0.250·16-s − 0.390·17-s + 0.923·19-s + 0.708·20-s + 1.00·22-s − 1.33·23-s + 1.00·25-s − 0.585·26-s − 1.53·29-s − 1.71·31-s − 0.176·32-s + 0.275·34-s − 0.717·37-s − 0.653·38-s − 0.500·40-s + 0.392·41-s − 1.16·43-s − 0.710·44-s + 0.944·46-s − 0.210·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 15.8T + 125T^{2} \) |
| 11 | \( 1 + 51.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 240.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 66.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 461.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 545.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 130.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 181.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 409.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 347.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
| 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
−9.479341125775286666430802948915, −8.590108869523462878104662957699, −7.73532615399993554119077673714, −6.81033209457823173796239785224, −5.69221429673130944695147216416, −5.38867215551372079608824636540, −3.62173075083196321389830684556, −2.34800380461761818977615785948, −1.61008240288151276559578093907, 0,
1.61008240288151276559578093907, 2.34800380461761818977615785948, 3.62173075083196321389830684556, 5.38867215551372079608824636540, 5.69221429673130944695147216416, 6.81033209457823173796239785224, 7.73532615399993554119077673714, 8.590108869523462878104662957699, 9.479341125775286666430802948915
