L(s) = 1 | + 76.9·3-s + 338.·5-s + 438.·7-s + 3.73e3·9-s + 1.96e3·11-s + 2.21e3·13-s + 2.60e4·15-s − 1.21e4·17-s + 3.28e4·19-s + 3.37e4·21-s − 1.96e4·23-s + 3.64e4·25-s + 1.19e5·27-s + 1.60e5·29-s − 2.29e5·31-s + 1.51e5·33-s + 1.48e5·35-s + 4.96e5·37-s + 1.70e5·39-s − 5.99e5·41-s − 8.83e4·43-s + 1.26e6·45-s + 8.20e5·47-s − 6.30e5·49-s − 9.32e5·51-s + 1.53e6·53-s + 6.65e5·55-s + ⋯ |
L(s) = 1 | + 1.64·3-s + 1.21·5-s + 0.483·7-s + 1.70·9-s + 0.445·11-s + 0.279·13-s + 1.99·15-s − 0.598·17-s + 1.09·19-s + 0.795·21-s − 0.335·23-s + 0.466·25-s + 1.16·27-s + 1.22·29-s − 1.38·31-s + 0.733·33-s + 0.585·35-s + 1.61·37-s + 0.459·39-s − 1.35·41-s − 0.169·43-s + 2.06·45-s + 1.15·47-s − 0.765·49-s − 0.984·51-s + 1.41·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(5.955423966\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.955423966\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 76.9T + 2.18e3T^{2} \) |
| 5 | \( 1 - 338.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 438.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.96e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.21e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.28e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.60e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.96e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.83e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 8.20e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.53e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.82e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.84e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.98e4T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.27e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.55e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.89e4T + 8.07e13T^{2} \) |
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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
−10.43064246784086935917712844034, −9.538537024918842311210646911878, −8.959580752256968093764760835509, −8.045881679337585051580468987806, −6.99960194537675909588012150951, −5.75242117588184067230339589591, −4.38567292015881479544929675241, −3.15276760577261726138176723117, −2.14490479486028363601341096421, −1.32195462733855491622249957308,
1.32195462733855491622249957308, 2.14490479486028363601341096421, 3.15276760577261726138176723117, 4.38567292015881479544929675241, 5.75242117588184067230339589591, 6.99960194537675909588012150951, 8.045881679337585051580468987806, 8.959580752256968093764760835509, 9.538537024918842311210646911878, 10.43064246784086935917712844034