| L(s) = 1 | − 4.48·2-s − 3.05·3-s + 12.0·4-s + 15.8·5-s + 13.6·6-s + 14.1·7-s − 18.3·8-s − 17.6·9-s − 71.0·10-s − 11·11-s − 36.9·12-s − 63.6·14-s − 48.4·15-s − 14.4·16-s − 5.15·17-s + 79.1·18-s + 53.7·19-s + 191.·20-s − 43.3·21-s + 49.3·22-s − 17.9·23-s + 56.1·24-s + 126.·25-s + 136.·27-s + 171.·28-s − 134.·29-s + 217.·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.587·3-s + 1.51·4-s + 1.41·5-s + 0.931·6-s + 0.766·7-s − 0.811·8-s − 0.654·9-s − 2.24·10-s − 0.301·11-s − 0.889·12-s − 1.21·14-s − 0.833·15-s − 0.225·16-s − 0.0735·17-s + 1.03·18-s + 0.649·19-s + 2.14·20-s − 0.450·21-s + 0.477·22-s − 0.162·23-s + 0.477·24-s + 1.00·25-s + 0.972·27-s + 1.15·28-s − 0.862·29-s + 1.32·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.48T + 8T^{2} \) |
| 3 | \( 1 + 3.05T + 27T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 17 | \( 1 + 5.15T + 4.91e3T^{2} \) |
| 19 | \( 1 - 53.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 17.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 43.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 27.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 165.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 542.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 655.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 969.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 174.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 53.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 841.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
| 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
−8.694301456747066407399192076110, −7.85139127117387397092521209795, −7.10059876955163562285674731535, −6.09414044742117494344048091759, −5.58253031895315692166379351782, −4.66684942791016465101367318171, −2.86009734879967599207534522284, −1.93964744894277708900593668943, −1.18331237775793858311077900873, 0,
1.18331237775793858311077900873, 1.93964744894277708900593668943, 2.86009734879967599207534522284, 4.66684942791016465101367318171, 5.58253031895315692166379351782, 6.09414044742117494344048091759, 7.10059876955163562285674731535, 7.85139127117387397092521209795, 8.694301456747066407399192076110
