| L(s) = 1 | − 5.35·2-s + 7.47·3-s + 20.7·4-s + 10.6·5-s − 40.0·6-s + 32.3·7-s − 68.0·8-s + 28.9·9-s − 57.3·10-s − 11·11-s + 154.·12-s − 173.·14-s + 79.9·15-s + 199.·16-s + 66.0·17-s − 154.·18-s + 126.·19-s + 221.·20-s + 241.·21-s + 58.9·22-s − 23.1·23-s − 508.·24-s − 10.5·25-s + 14.3·27-s + 669.·28-s + 174.·29-s − 428.·30-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.43·3-s + 2.58·4-s + 0.956·5-s − 2.72·6-s + 1.74·7-s − 3.00·8-s + 1.07·9-s − 1.81·10-s − 0.301·11-s + 3.72·12-s − 3.30·14-s + 1.37·15-s + 3.10·16-s + 0.942·17-s − 2.02·18-s + 1.52·19-s + 2.47·20-s + 2.51·21-s + 0.571·22-s − 0.209·23-s − 4.32·24-s − 0.0844·25-s + 0.102·27-s + 4.51·28-s + 1.11·29-s − 2.60·30-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
| \(L(2)\) |
\(\approx\) |
\(2.829003195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.829003195\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.35T + 8T^{2} \) |
| 3 | \( 1 - 7.47T + 27T^{2} \) |
| 5 | \( 1 - 10.6T + 125T^{2} \) |
| 7 | \( 1 - 32.3T + 343T^{2} \) |
| 17 | \( 1 - 66.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 105.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 77.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 219.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 29.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 161.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 16.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 508.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 532.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 708.T + 9.12e5T^{2} \) |
| 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
−8.804580420558081883749336913676, −8.120304025661701719436966592922, −7.86460771806976566946450819188, −7.09805386972224796130893089940, −5.91500589338041412424499371185, −4.93994295502600138135736363469, −3.24725078272625370306980570653, −2.44841519528461437907868834501, −1.66661691083039426360732867849, −1.09081132858999051273434388320,
1.09081132858999051273434388320, 1.66661691083039426360732867849, 2.44841519528461437907868834501, 3.24725078272625370306980570653, 4.93994295502600138135736363469, 5.91500589338041412424499371185, 7.09805386972224796130893089940, 7.86460771806976566946450819188, 8.120304025661701719436966592922, 8.804580420558081883749336913676
