| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 171.·5-s + 216·6-s − 343·7-s − 512·8-s + 729·9-s − 1.37e3·10-s + 1.69e3·11-s − 1.72e3·12-s − 2.19e3·13-s + 2.74e3·14-s − 4.63e3·15-s + 4.09e3·16-s − 2.09e4·17-s − 5.83e3·18-s + 1.04e4·19-s + 1.09e4·20-s + 9.26e3·21-s − 1.35e4·22-s + 1.21e4·23-s + 1.38e4·24-s − 4.86e4·25-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 + 0.614·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.434·10-s + 0.383·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.354·15-s + 0.250·16-s − 1.03·17-s − 0.235·18-s + 0.351·19-s + 0.307·20-s + 0.218·21-s − 0.271·22-s + 0.209·23-s + 0.204·24-s − 0.622·25-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
| \(L(4)\) |
\(\approx\) |
\(1.059453007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059453007\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 171.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 1.69e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.04e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.58e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.88e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.42e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.88e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.45e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.52e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 7.03e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.23e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.70e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.53e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.18e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.07e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.56e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.29e7T + 8.07e13T^{2} \) |
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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
−9.520887286052868547332362553449, −9.119449763276454698585408745054, −7.86503884318891897718537317949, −6.87974730195163858308603545757, −6.21203614321519186723498760103, −5.27655765727533216764293203414, −4.04243110814861964937895232133, −2.64971842708487274440968021067, −1.63199141630267021744800123430, −0.51399772406396517007318331501,
0.51399772406396517007318331501, 1.63199141630267021744800123430, 2.64971842708487274440968021067, 4.04243110814861964937895232133, 5.27655765727533216764293203414, 6.21203614321519186723498760103, 6.87974730195163858308603545757, 7.86503884318891897718537317949, 9.119449763276454698585408745054, 9.520887286052868547332362553449
