| L(s) = 1 | − 4.80·2-s + 3.56·3-s + 15.0·4-s − 13.7·5-s − 17.1·6-s + 7·7-s − 33.9·8-s − 14.3·9-s + 65.8·10-s − 19.4·11-s + 53.6·12-s − 0.895·13-s − 33.6·14-s − 48.8·15-s + 42.6·16-s + 68.7·18-s − 115.·19-s − 206.·20-s + 24.9·21-s + 93.3·22-s + 5.90·23-s − 121.·24-s + 63.1·25-s + 4.29·26-s − 147.·27-s + 105.·28-s − 0.926·29-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.685·3-s + 1.88·4-s − 1.22·5-s − 1.16·6-s + 0.377·7-s − 1.50·8-s − 0.530·9-s + 2.08·10-s − 0.532·11-s + 1.29·12-s − 0.0190·13-s − 0.641·14-s − 0.840·15-s + 0.666·16-s + 0.900·18-s − 1.39·19-s − 2.31·20-s + 0.259·21-s + 0.904·22-s + 0.0535·23-s − 1.02·24-s + 0.504·25-s + 0.0324·26-s − 1.04·27-s + 0.712·28-s − 0.00593·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2146264488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2146264488\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 4.80T + 8T^{2} \) |
| 3 | \( 1 - 3.56T + 27T^{2} \) |
| 5 | \( 1 + 13.7T + 125T^{2} \) |
| 11 | \( 1 + 19.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.895T + 2.19e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 5.90T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.926T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 386.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.77T + 6.89e4T^{2} \) |
| 43 | \( 1 + 264.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 615.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 1.72T + 2.05e5T^{2} \) |
| 61 | \( 1 - 111.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 986.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 116.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 73.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 176.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.663764940603904921849060890606, −8.221868297312917407267127486849, −7.58835606720088137277781544718, −7.01411741007170772359988887760, −5.88350291119080016079894879619, −4.58881757741531082395393563269, −3.55429511335304119679376280602, −2.58023541021990093252049097102, −1.68862458855550395073570555689, −0.25719252852957914072977825840,
0.25719252852957914072977825840, 1.68862458855550395073570555689, 2.58023541021990093252049097102, 3.55429511335304119679376280602, 4.58881757741531082395393563269, 5.88350291119080016079894879619, 7.01411741007170772359988887760, 7.58835606720088137277781544718, 8.221868297312917407267127486849, 8.663764940603904921849060890606
