| L(s) = 1 | + 2-s − 0.165·3-s + 4-s − 5-s − 0.165·6-s − 1.98·7-s + 8-s − 2.97·9-s − 10-s + 3.02·11-s − 0.165·12-s − 2.18·13-s − 1.98·14-s + 0.165·15-s + 16-s + 0.320·17-s − 2.97·18-s + 0.641·19-s − 20-s + 0.329·21-s + 3.02·22-s + 7.60·23-s − 0.165·24-s + 25-s − 2.18·26-s + 0.989·27-s − 1.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0956·3-s + 0.5·4-s − 0.447·5-s − 0.0676·6-s − 0.752·7-s + 0.353·8-s − 0.990·9-s − 0.316·10-s + 0.912·11-s − 0.0478·12-s − 0.606·13-s − 0.531·14-s + 0.0427·15-s + 0.250·16-s + 0.0777·17-s − 0.700·18-s + 0.147·19-s − 0.223·20-s + 0.0719·21-s + 0.645·22-s + 1.58·23-s − 0.0338·24-s + 0.200·25-s − 0.429·26-s + 0.190·27-s − 0.376·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.173588416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.173588416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 0.165T + 3T^{2} \) |
| 7 | \( 1 + 1.98T + 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 - 0.320T + 17T^{2} \) |
| 19 | \( 1 - 0.641T + 19T^{2} \) |
| 23 | \( 1 - 7.60T + 23T^{2} \) |
| 29 | \( 1 + 8.32T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 - 0.743T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 2.83T + 47T^{2} \) |
| 53 | \( 1 - 9.02T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 2.92T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 - 2.50T + 89T^{2} \) |
| 97 | \( 1 + 4.77T + 97T^{2} \) |
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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
−8.446415621378729653055080141233, −7.51472096259606720795927309948, −6.85046773257887345696165065997, −6.24121098185661351889580489986, −5.41521499180693436815024056534, −4.73236925323393253051710452196, −3.69856852839667271652836302301, −3.18872073913731719583776545121, −2.25923503093305810442766768408, −0.74644822843256720877476806236,
0.74644822843256720877476806236, 2.25923503093305810442766768408, 3.18872073913731719583776545121, 3.69856852839667271652836302301, 4.73236925323393253051710452196, 5.41521499180693436815024056534, 6.24121098185661351889580489986, 6.85046773257887345696165065997, 7.51472096259606720795927309948, 8.446415621378729653055080141233
