| L(s) = 1 | − 5·5-s + 30.4·7-s − 37.9·11-s − 43.3·13-s + 29.8·17-s + 28.6·19-s + 51.2·23-s + 25·25-s − 178.·29-s + 237.·31-s − 152.·35-s − 12.7·37-s − 222.·41-s − 464.·43-s + 400.·47-s + 586.·49-s − 249.·53-s + 189.·55-s − 779.·59-s + 609.·61-s + 216.·65-s + 172.·67-s − 1.04e3·71-s − 38.6·73-s − 1.15e3·77-s − 1.19e3·79-s + 150.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.64·7-s − 1.04·11-s − 0.925·13-s + 0.425·17-s + 0.345·19-s + 0.464·23-s + 0.200·25-s − 1.14·29-s + 1.37·31-s − 0.736·35-s − 0.0564·37-s − 0.848·41-s − 1.64·43-s + 1.24·47-s + 1.71·49-s − 0.646·53-s + 0.465·55-s − 1.71·59-s + 1.28·61-s + 0.414·65-s + 0.315·67-s − 1.74·71-s − 0.0619·73-s − 1.71·77-s − 1.70·79-s + 0.199·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 30.4T + 343T^{2} \) |
| 11 | \( 1 + 37.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 51.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 237.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 12.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 464.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 400.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 249.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 779.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 609.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 172.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 38.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 150.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 119.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−8.164469161423018861752560007304, −7.67047225044445807845785818649, −7.08050854175346862095510645867, −5.73537117788599248122141209537, −4.96926929817038081508935351691, −4.56304193048878833325781966818, −3.28231741178723094787648609750, −2.29964731430367246191271985723, −1.29226625441494257434469020694, 0,
1.29226625441494257434469020694, 2.29964731430367246191271985723, 3.28231741178723094787648609750, 4.56304193048878833325781966818, 4.96926929817038081508935351691, 5.73537117788599248122141209537, 7.08050854175346862095510645867, 7.67047225044445807845785818649, 8.164469161423018861752560007304
