| L(s) = 1 | − 4.22·2-s + 9.85·4-s − 5.85·5-s + 24.1·7-s − 7.85·8-s + 24.7·10-s − 33.8·11-s − 101.·14-s − 45.6·16-s + 49.3·17-s + 76.8·19-s − 57.7·20-s + 143.·22-s − 6.29·23-s − 90.6·25-s + 237.·28-s − 100.·29-s + 307.·31-s + 255.·32-s − 208.·34-s − 141.·35-s + 76.0·37-s − 324.·38-s + 46.0·40-s − 514.·41-s − 268.·43-s − 334.·44-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.23·4-s − 0.524·5-s + 1.30·7-s − 0.347·8-s + 0.783·10-s − 0.928·11-s − 1.94·14-s − 0.713·16-s + 0.704·17-s + 0.927·19-s − 0.645·20-s + 1.38·22-s − 0.0570·23-s − 0.725·25-s + 1.60·28-s − 0.646·29-s + 1.78·31-s + 1.41·32-s − 1.05·34-s − 0.682·35-s + 0.337·37-s − 1.38·38-s + 0.182·40-s − 1.95·41-s − 0.951·43-s − 1.14·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.22T + 8T^{2} \) |
| 5 | \( 1 + 5.85T + 125T^{2} \) |
| 7 | \( 1 - 24.1T + 343T^{2} \) |
| 11 | \( 1 + 33.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 49.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 76.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.29T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 514.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 67.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 25.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 588.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.00e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 895.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 968.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 119.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 16.6T + 9.12e5T^{2} \) |
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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
−8.397657165430214595242802067258, −8.035151104432918965706765694940, −7.63041296321229305743469761374, −6.63724092095745595669090911253, −5.30884709906757649436372859700, −4.65266403210223881873987067870, −3.28728943102379731908203741438, −2.02475394237595271426670357684, −1.14076464411841099358431151573, 0,
1.14076464411841099358431151573, 2.02475394237595271426670357684, 3.28728943102379731908203741438, 4.65266403210223881873987067870, 5.30884709906757649436372859700, 6.63724092095745595669090911253, 7.63041296321229305743469761374, 8.035151104432918965706765694940, 8.397657165430214595242802067258
