| L(s) = 1 | + 9.32·2-s + 54.9·4-s − 99.7·5-s + 125.·7-s + 214.·8-s − 930.·10-s − 177.·11-s − 919.·13-s + 1.17e3·14-s + 240.·16-s − 1.56e3·17-s + 447.·19-s − 5.48e3·20-s − 1.65e3·22-s + 529·23-s + 6.81e3·25-s − 8.57e3·26-s + 6.92e3·28-s + 1.29e3·29-s − 6.65e3·31-s − 4.61e3·32-s − 1.46e4·34-s − 1.25e4·35-s − 8.05e3·37-s + 4.17e3·38-s − 2.13e4·40-s − 1.79e4·41-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 1.71·4-s − 1.78·5-s + 0.971·7-s + 1.18·8-s − 2.94·10-s − 0.442·11-s − 1.50·13-s + 1.60·14-s + 0.235·16-s − 1.31·17-s + 0.284·19-s − 3.06·20-s − 0.730·22-s + 0.208·23-s + 2.18·25-s − 2.48·26-s + 1.67·28-s + 0.286·29-s − 1.24·31-s − 0.797·32-s − 2.17·34-s − 1.73·35-s − 0.967·37-s + 0.469·38-s − 2.11·40-s − 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 9.32T + 32T^{2} \) |
| 5 | \( 1 + 99.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 125.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 177.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 919.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.56e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 447.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 1.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.05e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.79e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.78e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.03e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.04e5T + 8.58e9T^{2} \) |
| 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
−11.45753341952795246274899398902, −10.70920981130083802671423911923, −8.741713999613928393615283364823, −7.56993171792478183357468199172, −6.96736703447452818538092760264, −5.16103936442287838475006853797, −4.60936165108928591535315987970, −3.63092596899130499872450297993, −2.33820534735904312861278639632, 0,
2.33820534735904312861278639632, 3.63092596899130499872450297993, 4.60936165108928591535315987970, 5.16103936442287838475006853797, 6.96736703447452818538092760264, 7.56993171792478183357468199172, 8.741713999613928393615283364823, 10.70920981130083802671423911923, 11.45753341952795246274899398902
