| L(s) = 1 | + 2.83·2-s + 3·3-s + 0.0563·4-s − 7.24·5-s + 8.51·6-s + 21.8·7-s − 22.5·8-s + 9·9-s − 20.5·10-s − 13.2·11-s + 0.169·12-s + 45.7·13-s + 62.1·14-s − 21.7·15-s − 64.4·16-s − 102.·17-s + 25.5·18-s + 86.3·19-s − 0.408·20-s + 65.6·21-s − 37.5·22-s − 23·23-s − 67.6·24-s − 72.4·25-s + 129.·26-s + 27·27-s + 1.23·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.577·3-s + 0.00704·4-s − 0.648·5-s + 0.579·6-s + 1.18·7-s − 0.996·8-s + 0.333·9-s − 0.650·10-s − 0.362·11-s + 0.00406·12-s + 0.976·13-s + 1.18·14-s − 0.374·15-s − 1.00·16-s − 1.45·17-s + 0.334·18-s + 1.04·19-s − 0.00456·20-s + 0.682·21-s − 0.364·22-s − 0.208·23-s − 0.575·24-s − 0.579·25-s + 0.979·26-s + 0.192·27-s + 0.00832·28-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
| \(L(2)\) |
\(\approx\) |
\(4.067031813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.067031813\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.83T + 8T^{2} \) |
| 5 | \( 1 + 7.24T + 125T^{2} \) |
| 7 | \( 1 - 21.8T + 343T^{2} \) |
| 11 | \( 1 + 13.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.3T + 6.85e3T^{2} \) |
| 31 | \( 1 - 291.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 98.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 85.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 280.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 106.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 346.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 20.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 716.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 985.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 561.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.570579261921114877076589275640, −8.165365102644122794561690937440, −7.29032653072537530385030984620, −6.27301607393014334074237238449, −5.38034100419497875199529931828, −4.48896043058905429607368187503, −4.09228162607428617053953510024, −3.11133390102165231257999606302, −2.14424099406605096758144897123, −0.78294274238745508379344788718,
0.78294274238745508379344788718, 2.14424099406605096758144897123, 3.11133390102165231257999606302, 4.09228162607428617053953510024, 4.48896043058905429607368187503, 5.38034100419497875199529931828, 6.27301607393014334074237238449, 7.29032653072537530385030984620, 8.165365102644122794561690937440, 8.570579261921114877076589275640
