| L(s) = 1 | − 4.73·2-s + 3·3-s + 14.4·4-s + 3.01·5-s − 14.2·6-s + 21.8·7-s − 30.5·8-s + 9·9-s − 14.2·10-s − 11·11-s + 43.3·12-s + 4.78·13-s − 103.·14-s + 9.03·15-s + 29.0·16-s + 125.·17-s − 42.6·18-s − 98.7·19-s + 43.4·20-s + 65.4·21-s + 52.1·22-s − 112.·23-s − 91.5·24-s − 115.·25-s − 22.6·26-s + 27·27-s + 315.·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.577·3-s + 1.80·4-s + 0.269·5-s − 0.966·6-s + 1.17·7-s − 1.34·8-s + 0.333·9-s − 0.451·10-s − 0.301·11-s + 1.04·12-s + 0.102·13-s − 1.97·14-s + 0.155·15-s + 0.453·16-s + 1.79·17-s − 0.558·18-s − 1.19·19-s + 0.486·20-s + 0.680·21-s + 0.504·22-s − 1.02·23-s − 0.778·24-s − 0.927·25-s − 0.171·26-s + 0.192·27-s + 2.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.486989539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.486989539\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 4.73T + 8T^{2} \) |
| 5 | \( 1 - 3.01T + 125T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 13 | \( 1 - 4.78T + 2.19e3T^{2} \) |
| 17 | \( 1 - 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 34.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 180.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 504.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 512.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 710.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 618.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 28.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 59.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 644.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 492.T + 9.12e5T^{2} \) |
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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
−8.560635803130550426704959962850, −8.089991963843965232272031814010, −7.81834464740390803439464611276, −6.78734126989090393065654682509, −5.84900695488498573385763422492, −4.78248875523803900091167358228, −3.60594151828321258620192166842, −2.28218841323138281556734326521, −1.73803608841201805888475457716, −0.72044639616434589191667938155,
0.72044639616434589191667938155, 1.73803608841201805888475457716, 2.28218841323138281556734326521, 3.60594151828321258620192166842, 4.78248875523803900091167358228, 5.84900695488498573385763422492, 6.78734126989090393065654682509, 7.81834464740390803439464611276, 8.089991963843965232272031814010, 8.560635803130550426704959962850
