| L(s) = 1 | − 4.83·2-s + 15.4·4-s + 21.1·5-s − 16.2·7-s − 35.8·8-s − 102.·10-s + 30.7·11-s + 78.7·14-s + 49.9·16-s − 46.2·17-s − 144.·19-s + 326.·20-s − 148.·22-s − 8.38·23-s + 324.·25-s − 250.·28-s + 242.·29-s − 87.9·31-s + 44.5·32-s + 223.·34-s − 345.·35-s + 49.6·37-s + 700.·38-s − 758.·40-s − 107.·41-s − 35.4·43-s + 473.·44-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.92·4-s + 1.89·5-s − 0.879·7-s − 1.58·8-s − 3.24·10-s + 0.842·11-s + 1.50·14-s + 0.781·16-s − 0.659·17-s − 1.74·19-s + 3.65·20-s − 1.44·22-s − 0.0759·23-s + 2.59·25-s − 1.69·28-s + 1.55·29-s − 0.509·31-s + 0.246·32-s + 1.12·34-s − 1.66·35-s + 0.220·37-s + 2.99·38-s − 3.00·40-s − 0.411·41-s − 0.125·43-s + 1.62·44-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 | \( 1 + 4.83T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 - 30.7T + 1.33e3T^{2} \) |
| 17 | \( 1 + 46.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.38T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 348.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 984.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 487.T + 9.12e5T^{2} \) |
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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
−8.981897085981772543749502106157, −8.314601621411837042982729123128, −6.89194185755863693949380545136, −6.49391500561329456011341224577, −6.00103191252237382150218888072, −4.56855563496647223679501156327, −2.91238929598446027942881519585, −2.06487948353137309191735914118, −1.31868120580712208546781031516, 0,
1.31868120580712208546781031516, 2.06487948353137309191735914118, 2.91238929598446027942881519585, 4.56855563496647223679501156327, 6.00103191252237382150218888072, 6.49391500561329456011341224577, 6.89194185755863693949380545136, 8.314601621411837042982729123128, 8.981897085981772543749502106157
