| L(s) = 1 | − 9·3-s + 46.6·5-s − 58.6·7-s + 81·9-s − 125.·11-s + 351.·13-s − 419.·15-s − 885.·17-s + 361·19-s + 527.·21-s + 936.·23-s − 952.·25-s − 729·27-s + 3.11e3·29-s − 1.05e4·31-s + 1.12e3·33-s − 2.73e3·35-s + 1.06e4·37-s − 3.15e3·39-s + 1.12e4·41-s + 5.84e3·43-s + 3.77e3·45-s − 4.84e3·47-s − 1.33e4·49-s + 7.97e3·51-s − 3.66e4·53-s − 5.85e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.833·5-s − 0.452·7-s + 0.333·9-s − 0.312·11-s + 0.576·13-s − 0.481·15-s − 0.743·17-s + 0.229·19-s + 0.261·21-s + 0.369·23-s − 0.304·25-s − 0.192·27-s + 0.688·29-s − 1.97·31-s + 0.180·33-s − 0.377·35-s + 1.27·37-s − 0.332·39-s + 1.04·41-s + 0.482·43-s + 0.277·45-s − 0.320·47-s − 0.795·49-s + 0.429·51-s − 1.79·53-s − 0.260·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 19 | \( 1 - 361T \) |
| good | 5 | \( 1 - 46.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 58.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 125.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 351.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 885.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 936.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.05e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.84e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.84e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.66e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.90e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.68e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.87e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.83e4T + 8.58e9T^{2} \) |
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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
−9.754933089230656438197049764550, −9.194943868555790324909401098876, −7.948449504769349789981212159808, −6.79398997538163434791779341229, −6.04064169888325815324651676423, −5.22945246005990428586326195485, −4.00064975841309095865402618046, −2.62599030347959456936313578465, −1.38672091519003482248555512758, 0,
1.38672091519003482248555512758, 2.62599030347959456936313578465, 4.00064975841309095865402618046, 5.22945246005990428586326195485, 6.04064169888325815324651676423, 6.79398997538163434791779341229, 7.948449504769349789981212159808, 9.194943868555790324909401098876, 9.754933089230656438197049764550
