| L(s) = 1 | + 2.11·2-s + 3·3-s − 3.50·4-s − 8.70·5-s + 6.35·6-s − 29.9·7-s − 24.3·8-s + 9·9-s − 18.4·10-s − 2.69·11-s − 10.5·12-s − 53.5·13-s − 63.5·14-s − 26.1·15-s − 23.6·16-s − 80.7·17-s + 19.0·18-s − 140.·19-s + 30.5·20-s − 89.8·21-s − 5.72·22-s + 23·23-s − 73.1·24-s − 49.1·25-s − 113.·26-s + 27·27-s + 105.·28-s + ⋯ |
| L(s) = 1 | + 0.749·2-s + 0.577·3-s − 0.438·4-s − 0.778·5-s + 0.432·6-s − 1.61·7-s − 1.07·8-s + 0.333·9-s − 0.583·10-s − 0.0739·11-s − 0.253·12-s − 1.14·13-s − 1.21·14-s − 0.449·15-s − 0.369·16-s − 1.15·17-s + 0.249·18-s − 1.69·19-s + 0.341·20-s − 0.933·21-s − 0.0554·22-s + 0.208·23-s − 0.622·24-s − 0.393·25-s − 0.855·26-s + 0.192·27-s + 0.709·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1483716666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1483716666\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 2.11T + 8T^{2} \) |
| 5 | \( 1 + 8.70T + 125T^{2} \) |
| 7 | \( 1 + 29.9T + 343T^{2} \) |
| 11 | \( 1 + 2.69T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 326.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 14.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 429.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 98.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 381.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 401.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 976.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 634.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 998.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 469.T + 9.12e5T^{2} \) |
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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
−8.909980518708764915561760050778, −8.083976880458237865302841429007, −7.06873031433906376283709070364, −6.48131066493999716974132449539, −5.54336659930485610811444488246, −4.31663436464654502780796697360, −4.05156906325453741010985424895, −3.05129640414016610679103225448, −2.34081987279852848228334919944, −0.14110470992689642411742920156,
0.14110470992689642411742920156, 2.34081987279852848228334919944, 3.05129640414016610679103225448, 4.05156906325453741010985424895, 4.31663436464654502780796697360, 5.54336659930485610811444488246, 6.48131066493999716974132449539, 7.06873031433906376283709070364, 8.083976880458237865302841429007, 8.909980518708764915561760050778
