| L(s) = 1 | + 5.23·2-s − 100.·4-s + 769.·7-s − 1.19e3·8-s − 5.35e3·11-s + 1.40e4·13-s + 4.03e3·14-s + 6.60e3·16-s − 1.24e4·17-s − 5.03e3·19-s − 2.80e4·22-s + 7.51e4·23-s + 7.35e4·26-s − 7.74e4·28-s − 1.95e5·29-s − 9.35e4·31-s + 1.87e5·32-s − 6.49e4·34-s + 1.61e5·37-s − 2.63e4·38-s + 2.87e4·41-s + 7.39e5·43-s + 5.38e5·44-s + 3.93e5·46-s − 1.06e6·47-s − 2.30e5·49-s − 1.41e6·52-s + ⋯ |
| L(s) = 1 | + 0.462·2-s − 0.785·4-s + 0.848·7-s − 0.826·8-s − 1.21·11-s + 1.77·13-s + 0.392·14-s + 0.402·16-s − 0.612·17-s − 0.168·19-s − 0.561·22-s + 1.28·23-s + 0.820·26-s − 0.666·28-s − 1.48·29-s − 0.564·31-s + 1.01·32-s − 0.283·34-s + 0.524·37-s − 0.0780·38-s + 0.0651·41-s + 1.41·43-s + 0.953·44-s + 0.596·46-s − 1.50·47-s − 0.280·49-s − 1.39·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.23T + 128T^{2} \) |
| 7 | \( 1 - 769.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.40e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.24e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.03e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.51e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.06e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.26e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.57e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.74e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.01e7T + 8.07e13T^{2} \) |
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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
−10.82240474808395169206346356059, −9.330578717960758147268469799069, −8.546911209444174447563986919540, −7.66706733940980461384285652748, −6.08619383958503047758826069483, −5.18460701474627333684445191402, −4.24081727918334304586740643913, −3.06493819291444325398284792649, −1.43798087036939164292888391780, 0,
1.43798087036939164292888391780, 3.06493819291444325398284792649, 4.24081727918334304586740643913, 5.18460701474627333684445191402, 6.08619383958503047758826069483, 7.66706733940980461384285652748, 8.546911209444174447563986919540, 9.330578717960758147268469799069, 10.82240474808395169206346356059
