| L(s) = 1 | + 73.9·3-s − 358.·5-s − 109.·7-s + 3.27e3·9-s − 7.15e3·11-s − 3.12e3·13-s − 2.65e4·15-s − 2.38e4·17-s − 6.85e3·19-s − 8.11e3·21-s + 5.06e4·23-s + 5.07e4·25-s + 8.03e4·27-s − 1.34e5·29-s + 1.99e5·31-s − 5.28e5·33-s + 3.94e4·35-s − 9.54e4·37-s − 2.30e5·39-s − 4.88e5·41-s + 1.62e5·43-s − 1.17e6·45-s − 8.32e5·47-s − 8.11e5·49-s − 1.76e6·51-s + 9.53e5·53-s + 2.56e6·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s − 1.28·5-s − 0.120·7-s + 1.49·9-s − 1.62·11-s − 0.394·13-s − 2.02·15-s − 1.17·17-s − 0.229·19-s − 0.191·21-s + 0.867·23-s + 0.649·25-s + 0.785·27-s − 1.02·29-s + 1.20·31-s − 2.56·33-s + 0.155·35-s − 0.309·37-s − 0.623·39-s − 1.10·41-s + 0.312·43-s − 1.92·45-s − 1.16·47-s − 0.985·49-s − 1.86·51-s + 0.879·53-s + 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 - 73.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 358.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 109.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.15e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.12e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.38e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 5.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.99e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.54e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.88e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.32e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.53e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.46e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.59e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.90e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.17e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.26e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.80e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.56e6T + 8.07e13T^{2} \) |
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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
−12.87645839178033409564207417027, −11.41175334672002421337414684915, −10.13180835275794533788599683687, −8.754923715995438851696599823015, −8.013659569154647037090963357717, −7.14087023433922310854911057027, −4.72945928646603151656852285482, −3.42736479409068320434349872190, −2.36430134544470230845786203419, 0,
2.36430134544470230845786203419, 3.42736479409068320434349872190, 4.72945928646603151656852285482, 7.14087023433922310854911057027, 8.013659569154647037090963357717, 8.754923715995438851696599823015, 10.13180835275794533788599683687, 11.41175334672002421337414684915, 12.87645839178033409564207417027
