L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 516.·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 4.13e3·10-s − 1.47e3·11-s + 1.72e3·12-s + 2.19e3·13-s − 2.74e3·14-s − 1.39e4·15-s + 4.09e3·16-s + 1.45e4·17-s + 5.83e3·18-s + 1.68e4·19-s − 3.30e4·20-s − 9.26e3·21-s − 1.17e4·22-s + 6.48e4·23-s + 1.38e4·24-s + 1.88e5·25-s + 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.84·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.30·10-s − 0.333·11-s + 0.288·12-s + 0.277·13-s − 0.267·14-s − 1.06·15-s + 0.250·16-s + 0.718·17-s + 0.235·18-s + 0.564·19-s − 0.923·20-s − 0.218·21-s − 0.235·22-s + 1.11·23-s + 0.204·24-s + 2.41·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 5 | \( 1 + 516.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 1.47e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.23e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.21e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.52e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.39e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.91e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.31e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.89e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.55e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 5.94e6T + 8.07e13T^{2} \) |
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
−9.040212484041951801600950000475, −8.108034760016223827127449384811, −7.47492244394749194675770620516, −6.71337277206907552260292489329, −5.27029241939908995757640831361, −4.34466889432099229667444067442, −3.40465492227486525421696827968, −2.98486393280944505775422126308, −1.25071996214478156010945813700, 0,
1.25071996214478156010945813700, 2.98486393280944505775422126308, 3.40465492227486525421696827968, 4.34466889432099229667444067442, 5.27029241939908995757640831361, 6.71337277206907552260292489329, 7.47492244394749194675770620516, 8.108034760016223827127449384811, 9.040212484041951801600950000475