| L(s) = 1 | + 27·3-s − 338.·5-s + 1.29e3·7-s + 729·9-s − 3.71e3·11-s + 1.34e4·13-s − 9.14e3·15-s + 2.65e3·17-s − 2.97e4·19-s + 3.48e4·21-s − 5.18e4·23-s + 3.66e4·25-s + 1.96e4·27-s − 1.74e5·29-s + 3.16e5·31-s − 1.00e5·33-s − 4.37e5·35-s + 3.30e5·37-s + 3.63e5·39-s + 3.32e5·41-s + 3.45e5·43-s − 2.46e5·45-s + 1.03e6·47-s + 8.43e5·49-s + 7.17e4·51-s + 1.09e6·53-s + 1.25e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.21·5-s + 1.42·7-s + 0.333·9-s − 0.841·11-s + 1.70·13-s − 0.699·15-s + 0.131·17-s − 0.993·19-s + 0.821·21-s − 0.888·23-s + 0.468·25-s + 0.192·27-s − 1.33·29-s + 1.90·31-s − 0.486·33-s − 1.72·35-s + 1.07·37-s + 0.982·39-s + 0.753·41-s + 0.662·43-s − 0.403·45-s + 1.45·47-s + 1.02·49-s + 0.0757·51-s + 1.00·53-s + 1.02·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.481249602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481249602\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 + 338.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.29e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.71e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.34e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.65e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.18e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.74e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.16e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.30e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.32e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.45e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.03e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.09e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.45e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.00e7T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.59e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.07e6T + 8.07e13T^{2} \) |
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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
−11.18697487444311519681818449431, −10.56216207128243916201539619048, −8.885255814100418532569973627564, −8.027477702398615439093171454229, −7.72514770200717673768764197173, −6.01447388519959093139953395863, −4.51158849316578598741223927194, −3.78318757839233316665609002753, −2.24001058082147826397627801720, −0.850426706371293927609897951279,
0.850426706371293927609897951279, 2.24001058082147826397627801720, 3.78318757839233316665609002753, 4.51158849316578598741223927194, 6.01447388519959093139953395863, 7.72514770200717673768764197173, 8.027477702398615439093171454229, 8.885255814100418532569973627564, 10.56216207128243916201539619048, 11.18697487444311519681818449431
