| L(s) = 1 | + 27·3-s + 530.·5-s + 930.·7-s + 729·9-s + 4.80e3·11-s − 1.02e4·13-s + 1.43e4·15-s − 2.56e4·17-s + 2.44e4·19-s + 2.51e4·21-s + 1.95e4·23-s + 2.03e5·25-s + 1.96e4·27-s + 2.42e5·29-s − 1.37e5·31-s + 1.29e5·33-s + 4.93e5·35-s − 1.46e5·37-s − 2.75e5·39-s + 2.79e5·41-s + 7.93e5·43-s + 3.86e5·45-s − 1.15e6·47-s + 4.19e4·49-s − 6.93e5·51-s + 6.81e5·53-s + 2.54e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.89·5-s + 1.02·7-s + 0.333·9-s + 1.08·11-s − 1.28·13-s + 1.09·15-s − 1.26·17-s + 0.817·19-s + 0.591·21-s + 0.334·23-s + 2.60·25-s + 0.192·27-s + 1.84·29-s − 0.831·31-s + 0.628·33-s + 1.94·35-s − 0.476·37-s − 0.744·39-s + 0.634·41-s + 1.52·43-s + 0.632·45-s − 1.61·47-s + 0.0509·49-s − 0.731·51-s + 0.628·53-s + 2.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.240714937\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.240714937\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 530.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 930.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.80e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.02e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.44e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.42e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.79e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.93e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.15e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.00e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.79e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.69e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.24e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.63e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.23e6T + 8.07e13T^{2} \) |
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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
−9.930696936645783772891875907315, −9.273958407948690429419814565657, −8.598773317958026513111363935719, −7.23863727001184823005031672908, −6.42744691369389915850605404262, −5.25070925563246375859317594200, −4.49043105620446623285615228701, −2.75343699660312644344482331652, −1.97304497935835854052828255727, −1.14108510848980635296922093636,
1.14108510848980635296922093636, 1.97304497935835854052828255727, 2.75343699660312644344482331652, 4.49043105620446623285615228701, 5.25070925563246375859317594200, 6.42744691369389915850605404262, 7.23863727001184823005031672908, 8.598773317958026513111363935719, 9.273958407948690429419814565657, 9.930696936645783772891875907315
