L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 223.·5-s − 216·6-s − 343·7-s + 512·8-s + 729·9-s − 1.78e3·10-s + 3.70e3·11-s − 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 6.03e3·15-s + 4.09e3·16-s − 5.32e3·17-s + 5.83e3·18-s − 847.·19-s − 1.43e4·20-s + 9.26e3·21-s + 2.96e4·22-s − 6.77e3·23-s − 1.38e4·24-s − 2.81e4·25-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 − 0.799·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.565·10-s + 0.839·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.461·15-s + 0.250·16-s − 0.262·17-s + 0.235·18-s − 0.0283·19-s − 0.399·20-s + 0.218·21-s + 0.593·22-s − 0.116·23-s − 0.204·24-s − 0.360·25-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 | \( 1 + 223.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.70e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 5.32e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 847.T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.77e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.14e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.00e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.73e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.76e3T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.33e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.46e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.08e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.47e6T + 8.07e13T^{2} \) |
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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
−9.333619747144469265323517457097, −8.149321651234494189373438918677, −7.17775751799425529911223177712, −6.44685819444608689812653517529, −5.53096052077554680512234583123, −4.38206914783492752007652907002, −3.81046221169447953713417846572, −2.57176978711130033645285643436, −1.16268827239207700012664434249, 0,
1.16268827239207700012664434249, 2.57176978711130033645285643436, 3.81046221169447953713417846572, 4.38206914783492752007652907002, 5.53096052077554680512234583123, 6.44685819444608689812653517529, 7.17775751799425529911223177712, 8.149321651234494189373438918677, 9.333619747144469265323517457097