| L(s) = 1 | − 8.95·2-s − 21.0·3-s − 47.7·4-s − 55.0·5-s + 188.·6-s − 343·7-s + 1.57e3·8-s − 1.74e3·9-s + 492.·10-s − 6.31e3·11-s + 1.00e3·12-s − 2.19e3·13-s + 3.07e3·14-s + 1.15e3·15-s − 7.99e3·16-s − 3.28e4·17-s + 1.56e4·18-s − 3.99e4·19-s + 2.62e3·20-s + 7.22e3·21-s + 5.65e4·22-s + 9.03e4·23-s − 3.31e4·24-s − 7.50e4·25-s + 1.96e4·26-s + 8.28e4·27-s + 1.63e4·28-s + ⋯ |
| L(s) = 1 | − 0.791·2-s − 0.450·3-s − 0.372·4-s − 0.196·5-s + 0.356·6-s − 0.377·7-s + 1.08·8-s − 0.796·9-s + 0.155·10-s − 1.43·11-s + 0.168·12-s − 0.277·13-s + 0.299·14-s + 0.0886·15-s − 0.488·16-s − 1.62·17-s + 0.631·18-s − 1.33·19-s + 0.0733·20-s + 0.170·21-s + 1.13·22-s + 1.54·23-s − 0.489·24-s − 0.961·25-s + 0.219·26-s + 0.809·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1762703767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1762703767\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 8.95T + 128T^{2} \) |
| 3 | \( 1 + 21.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 55.0T + 7.81e4T^{2} \) |
| 11 | \( 1 + 6.31e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.28e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.21e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.12e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.95e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.02e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.89e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.15e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.03e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.15e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 7.89e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.59e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.59e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.89e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.32e6T + 8.07e13T^{2} \) |
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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
−12.84473810392411878745393976369, −11.17166707399635766759327981322, −10.59628136558821095968996692583, −9.250467645595288165849028456179, −8.393085313023955982247851115421, −7.19725456493668094058657335552, −5.64835831033837951420184212333, −4.39362801141996217738274098996, −2.42923433546159347910989709052, −0.28917015925136282964672553835,
0.28917015925136282964672553835, 2.42923433546159347910989709052, 4.39362801141996217738274098996, 5.64835831033837951420184212333, 7.19725456493668094058657335552, 8.393085313023955982247851115421, 9.250467645595288165849028456179, 10.59628136558821095968996692583, 11.17166707399635766759327981322, 12.84473810392411878745393976369
