L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 125·5-s − 216·6-s − 343·7-s − 512·8-s + 729·9-s + 1.00e3·10-s + 2.54e3·11-s + 1.72e3·12-s − 8.77e3·13-s + 2.74e3·14-s − 3.37e3·15-s + 4.09e3·16-s + 1.81e4·17-s − 5.83e3·18-s + 4.40e4·19-s − 8.00e3·20-s − 9.26e3·21-s − 2.03e4·22-s − 7.48e4·23-s − 1.38e4·24-s + 1.56e4·25-s + 7.01e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.575·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s − 0.258·15-s + 0.250·16-s + 0.893·17-s − 0.235·18-s + 1.47·19-s − 0.223·20-s − 0.218·21-s − 0.407·22-s − 1.28·23-s − 0.204·24-s + 0.199·25-s + 0.783·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 27T \) |
| 5 | \( 1 + 125T \) |
| 7 | \( 1 + 343T \) |
good | 11 | \( 1 - 2.54e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.77e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.81e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.17e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.42e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.67e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.59e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.67e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.69e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.61e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.12e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.65e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.92e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.37e4T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.41e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16299929265209424163904142619, −9.720730161872555437167027423018, −8.616197762131426286869676739047, −7.65008675446211408163802603640, −6.92942446448054310847997833729, −5.44645405450373176186635407216, −3.85918793417774794486222491410, −2.80280750914621697403768538850, −1.39525929571679094206715486959, 0,
1.39525929571679094206715486959, 2.80280750914621697403768538850, 3.85918793417774794486222491410, 5.44645405450373176186635407216, 6.92942446448054310847997833729, 7.65008675446211408163802603640, 8.616197762131426286869676739047, 9.720730161872555437167027423018, 10.16299929265209424163904142619