| L(s) = 1 | + 2.63·2-s + 4.02·3-s − 1.04·4-s − 16.9·5-s + 10.6·6-s − 23.8·8-s − 10.8·9-s − 44.8·10-s − 45.8·11-s − 4.18·12-s + 12.4·13-s − 68.3·15-s − 54.5·16-s − 28.8·17-s − 28.4·18-s − 134.·19-s + 17.6·20-s − 121.·22-s + 125.·23-s − 95.9·24-s + 163.·25-s + 32.7·26-s − 152.·27-s − 130.·29-s − 180.·30-s − 77.9·31-s + 46.7·32-s + ⋯ |
| L(s) = 1 | + 0.932·2-s + 0.774·3-s − 0.130·4-s − 1.51·5-s + 0.722·6-s − 1.05·8-s − 0.400·9-s − 1.41·10-s − 1.25·11-s − 0.100·12-s + 0.264·13-s − 1.17·15-s − 0.852·16-s − 0.411·17-s − 0.373·18-s − 1.62·19-s + 0.197·20-s − 1.17·22-s + 1.13·23-s − 0.816·24-s + 1.31·25-s + 0.246·26-s − 1.08·27-s − 0.838·29-s − 1.09·30-s − 0.451·31-s + 0.258·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7683416042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7683416042\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 2.63T + 8T^{2} \) |
| 3 | \( 1 - 4.02T + 27T^{2} \) |
| 5 | \( 1 + 16.9T + 125T^{2} \) |
| 11 | \( 1 + 45.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 130.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 299.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 392.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 283.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 27.5T + 3.89e5T^{2} \) |
| 79 | \( 1 + 424.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 167.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 288.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 415.T + 9.12e5T^{2} \) |
| 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
−8.555105966455739707972171174629, −8.018333870298616668435208244046, −7.22546193189646392054535212447, −6.20767235715812161793260574582, −5.22884983082514742308018729125, −4.49698464620392452195087544912, −3.76262616375097628008190408598, −3.12229725983104707096966246868, −2.31608732859296194623875526221, −0.30352105550147122602469332614,
0.30352105550147122602469332614, 2.31608732859296194623875526221, 3.12229725983104707096966246868, 3.76262616375097628008190408598, 4.49698464620392452195087544912, 5.22884983082514742308018729125, 6.20767235715812161793260574582, 7.22546193189646392054535212447, 8.018333870298616668435208244046, 8.555105966455739707972171174629
