| L(s) = 1 | − 5·5-s − 26.8·7-s + 30.8·11-s + 32.8·13-s − 71.5·17-s + 49.3·19-s + 72.5·23-s + 25·25-s − 54.4·29-s − 146.·31-s + 134.·35-s − 65.6·37-s − 148.·41-s + 453.·43-s − 171.·47-s + 376.·49-s + 440.·53-s − 154.·55-s − 128.·59-s + 395.·61-s − 164.·65-s + 380.·67-s + 490.·71-s − 288.·73-s − 827.·77-s + 395.·79-s + 845.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.44·7-s + 0.845·11-s + 0.701·13-s − 1.02·17-s + 0.595·19-s + 0.657·23-s + 0.200·25-s − 0.348·29-s − 0.850·31-s + 0.647·35-s − 0.291·37-s − 0.566·41-s + 1.60·43-s − 0.531·47-s + 1.09·49-s + 1.14·53-s − 0.378·55-s − 0.282·59-s + 0.830·61-s − 0.313·65-s + 0.694·67-s + 0.819·71-s − 0.462·73-s − 1.22·77-s + 0.563·79-s + 1.11·83-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 | \( 1 + 26.8T + 343T^{2} \) |
| 11 | \( 1 - 30.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 49.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 72.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 54.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 65.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 453.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 171.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 440.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 128.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 395.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 380.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 490.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 288.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 395.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 845.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 743.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.84e3T + 9.12e5T^{2} \) |
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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
−8.513062970613495247940250224391, −7.37301312236406442815372733888, −6.74901226599344609439349845533, −6.16333053197700728864454295621, −5.17541411240472009708644334159, −3.94006085508132221144180036284, −3.54965535696964745841732183061, −2.47308569824954038924268770288, −1.08476778333450002225523145844, 0,
1.08476778333450002225523145844, 2.47308569824954038924268770288, 3.54965535696964745841732183061, 3.94006085508132221144180036284, 5.17541411240472009708644334159, 6.16333053197700728864454295621, 6.74901226599344609439349845533, 7.37301312236406442815372733888, 8.513062970613495247940250224391
