| L(s) = 1 | − 0.654·2-s − 7.57·4-s − 5·5-s + 3.49·7-s + 10.1·8-s + 3.27·10-s + 11·11-s + 23.2·13-s − 2.28·14-s + 53.9·16-s − 23.2·17-s − 28.9·19-s + 37.8·20-s − 7.19·22-s + 76.2·23-s + 25·25-s − 15.2·26-s − 26.4·28-s + 162.·29-s − 114.·31-s − 116.·32-s + 15.1·34-s − 17.4·35-s − 271.·37-s + 18.9·38-s − 50.9·40-s + 5.16·41-s + ⋯ |
| L(s) = 1 | − 0.231·2-s − 0.946·4-s − 0.447·5-s + 0.188·7-s + 0.450·8-s + 0.103·10-s + 0.301·11-s + 0.495·13-s − 0.0436·14-s + 0.842·16-s − 0.331·17-s − 0.350·19-s + 0.423·20-s − 0.0697·22-s + 0.690·23-s + 0.200·25-s − 0.114·26-s − 0.178·28-s + 1.04·29-s − 0.661·31-s − 0.645·32-s + 0.0766·34-s − 0.0844·35-s − 1.20·37-s + 0.0809·38-s − 0.201·40-s + 0.0196·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 0.654T + 8T^{2} \) |
| 7 | \( 1 - 3.49T + 343T^{2} \) |
| 13 | \( 1 - 23.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 76.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 5.16T + 6.89e4T^{2} \) |
| 43 | \( 1 + 98.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 7.88T + 1.48e5T^{2} \) |
| 59 | \( 1 - 255.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 591.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 331.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 63.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 596.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 117.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 907.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 40.7T + 9.12e5T^{2} \) |
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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
−10.07303157558769343685435368571, −8.976384241973795108568573932483, −8.514508335863953567062354086510, −7.52078498978475109217123898556, −6.42877691658476607200351798051, −5.14858940212960091225403748867, −4.30702523537250071947131702466, −3.26822692992018994838605482352, −1.40580226694186892525287798465, 0,
1.40580226694186892525287798465, 3.26822692992018994838605482352, 4.30702523537250071947131702466, 5.14858940212960091225403748867, 6.42877691658476607200351798051, 7.52078498978475109217123898556, 8.514508335863953567062354086510, 8.976384241973795108568573932483, 10.07303157558769343685435368571
