| L(s) = 1 | + 9·3-s − 19.5·5-s − 97.9·7-s + 81·9-s + 142·11-s − 176.·15-s − 881.·21-s − 241.·25-s + 729·27-s + 1.46e3·29-s + 1.86e3·31-s + 1.27e3·33-s + 1.91e3·35-s − 1.58e3·45-s + 7.19e3·49-s + 4.60e3·53-s − 2.78e3·55-s − 6.86e3·59-s − 7.93e3·63-s + 8.15e3·73-s − 2.16e3·75-s − 1.39e4·77-s + 8.52e3·79-s + 6.56e3·81-s − 4.17e3·83-s + 1.32e4·87-s + 1.67e4·93-s + ⋯ |
| L(s) = 1 | + 3-s − 0.783·5-s − 1.99·7-s + 9-s + 1.17·11-s − 0.783·15-s − 1.99·21-s − 0.385·25-s + 0.999·27-s + 1.74·29-s + 1.93·31-s + 1.17·33-s + 1.56·35-s − 0.783·45-s + 2.99·49-s + 1.63·53-s − 0.919·55-s − 1.97·59-s − 1.99·63-s + 1.53·73-s − 0.385·75-s − 2.34·77-s + 1.36·79-s + 81-s − 0.606·83-s + 1.74·87-s + 1.93·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.092652702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.092652702\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| good | 5 | \( 1 + 19.5T + 625T^{2} \) |
| 7 | \( 1 + 97.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 142T + 1.46e4T^{2} \) |
| 13 | \( 1 - 2.85e4T^{2} \) |
| 17 | \( 1 - 8.35e4T^{2} \) |
| 19 | \( 1 - 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.46e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.86e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.60e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 6.86e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 1.38e7T^{2} \) |
| 67 | \( 1 - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.15e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.52e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.17e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.72e4T + 8.85e7T^{2} \) |
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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
−10.39220456587035973264790395994, −9.675016392611588363553641865439, −8.954051723152873209420788442368, −8.017948514145468185536971421514, −6.85568308932687972245162355129, −6.33871604860404353897731076579, −4.33774680984265577160735086745, −3.54750103536048768627815563457, −2.70645891677297904096548516511, −0.808910801807589033063655189094,
0.808910801807589033063655189094, 2.70645891677297904096548516511, 3.54750103536048768627815563457, 4.33774680984265577160735086745, 6.33871604860404353897731076579, 6.85568308932687972245162355129, 8.017948514145468185536971421514, 8.954051723152873209420788442368, 9.675016392611588363553641865439, 10.39220456587035973264790395994
