| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 292.·5-s − 216·6-s + 343·7-s − 512·8-s + 729·9-s + 2.34e3·10-s − 186.·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 7.90e3·15-s + 4.09e3·16-s + 3.66e4·17-s − 5.83e3·18-s − 8.59e3·19-s − 1.87e4·20-s + 9.26e3·21-s + 1.49e3·22-s − 1.27e4·23-s − 1.38e4·24-s + 7.61e3·25-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 − 1.04·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.740·10-s − 0.0422·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.604·15-s + 0.250·16-s + 1.81·17-s − 0.235·18-s − 0.287·19-s − 0.523·20-s + 0.218·21-s + 0.0298·22-s − 0.218·23-s − 0.204·24-s + 0.0975·25-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
| \(L(4)\) |
\(\approx\) |
\(1.542140639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.542140639\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 292.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 186.T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 8.59e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.25e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.11e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.15e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.55e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.95e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.34e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.12e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.51e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.50e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.46e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.84e6T + 8.07e13T^{2} \) |
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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
−9.637969471580303135300960747156, −8.629403954162668287469589167191, −7.79107193635865590337807210053, −7.56336718635509749580567305856, −6.24420056480028743654802320643, −4.94373685227357781663222178135, −3.79533309585980862175661521209, −2.94243314687110265348178127161, −1.66402966457206636809137788579, −0.59265922906405197684288496811,
0.59265922906405197684288496811, 1.66402966457206636809137788579, 2.94243314687110265348178127161, 3.79533309585980862175661521209, 4.94373685227357781663222178135, 6.24420056480028743654802320643, 7.56336718635509749580567305856, 7.79107193635865590337807210053, 8.629403954162668287469589167191, 9.637969471580303135300960747156
