| L(s) = 1 | − 11.5·2-s + 5.14·4-s + 125·5-s − 343·7-s + 1.41e3·8-s − 1.44e3·10-s − 8.58e3·11-s − 7.49e3·13-s + 3.95e3·14-s − 1.70e4·16-s + 1.91e4·17-s − 5.84e4·19-s + 642.·20-s + 9.90e4·22-s + 3.96e4·23-s + 1.56e4·25-s + 8.64e4·26-s − 1.76e3·28-s + 1.67e5·29-s − 2.11e5·31-s + 1.48e4·32-s − 2.21e5·34-s − 4.28e4·35-s − 2.59e5·37-s + 6.74e5·38-s + 1.77e5·40-s − 6.06e5·41-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.0401·4-s + 0.447·5-s − 0.377·7-s + 0.978·8-s − 0.456·10-s − 1.94·11-s − 0.946·13-s + 0.385·14-s − 1.03·16-s + 0.945·17-s − 1.95·19-s + 0.0179·20-s + 1.98·22-s + 0.678·23-s + 0.199·25-s + 0.964·26-s − 0.0151·28-s + 1.27·29-s − 1.27·31-s + 0.0802·32-s − 0.964·34-s − 0.169·35-s − 0.843·37-s + 1.99·38-s + 0.437·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3665327114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3665327114\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 125T \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 11.5T + 128T^{2} \) |
| 11 | \( 1 + 8.58e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.49e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.91e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.84e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.67e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.11e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.59e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.06e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.10e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.37e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.70e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.71e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.96e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.75e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.39e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.35e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.40e6T + 8.07e13T^{2} \) |
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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
−10.42703027186526179381261887352, −9.575674762271460752183372783806, −8.580650000177441409331406314443, −7.81883830324803483271271893953, −6.88777741653067880410313849145, −5.47932624234387837268007658377, −4.62233869744241909775488058371, −2.90073084993387940017573125007, −1.85493690299370276416057422499, −0.32995984817426760118821690634,
0.32995984817426760118821690634, 1.85493690299370276416057422499, 2.90073084993387940017573125007, 4.62233869744241909775488058371, 5.47932624234387837268007658377, 6.88777741653067880410313849145, 7.81883830324803483271271893953, 8.580650000177441409331406314443, 9.575674762271460752183372783806, 10.42703027186526179381261887352
