| L(s) = 1 | + 9.55·3-s − 14.9·5-s + 16.3·7-s + 64.3·9-s + 46.9·11-s − 21.8·13-s − 142.·15-s − 8.65·17-s + 34.4·19-s + 156.·21-s − 78.5·23-s + 97.9·25-s + 356.·27-s + 99.1·29-s + 26.9·31-s + 448.·33-s − 244.·35-s + 445.·37-s − 208.·39-s + 186.·41-s − 392.·43-s − 960.·45-s + 214.·47-s − 75.8·49-s − 82.7·51-s + 368.·53-s − 700.·55-s + ⋯ |
| L(s) = 1 | + 1.83·3-s − 1.33·5-s + 0.882·7-s + 2.38·9-s + 1.28·11-s − 0.465·13-s − 2.45·15-s − 0.123·17-s + 0.415·19-s + 1.62·21-s − 0.712·23-s + 0.783·25-s + 2.54·27-s + 0.634·29-s + 0.156·31-s + 2.36·33-s − 1.17·35-s + 1.97·37-s − 0.856·39-s + 0.709·41-s − 1.39·43-s − 3.18·45-s + 0.665·47-s − 0.221·49-s − 0.227·51-s + 0.955·53-s − 1.71·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.162789466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.162789466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 9.55T + 27T^{2} \) |
| 5 | \( 1 + 14.9T + 125T^{2} \) |
| 7 | \( 1 - 16.3T + 343T^{2} \) |
| 11 | \( 1 - 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.65T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 78.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 26.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 445.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 392.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 214.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 60.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 803.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 303.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 293.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 383.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 179.T + 9.12e5T^{2} \) |
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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
−9.290522546183171554875977545296, −8.636110305182609329266624516095, −7.83491655022356747533078735774, −7.59923801722725333111569774737, −6.51387800491526360578136436167, −4.66148908005816105871947901043, −4.09868260185307833647759611663, −3.33302416673510407023766453153, −2.22660066443784377891790677477, −1.07037853257089919400714244393,
1.07037853257089919400714244393, 2.22660066443784377891790677477, 3.33302416673510407023766453153, 4.09868260185307833647759611663, 4.66148908005816105871947901043, 6.51387800491526360578136436167, 7.59923801722725333111569774737, 7.83491655022356747533078735774, 8.636110305182609329266624516095, 9.290522546183171554875977545296
