| L(s) = 1 | + 1.64·2-s − 0.819·3-s + 0.695·4-s + 3.58·5-s − 1.34·6-s − 2.14·8-s − 2.32·9-s + 5.89·10-s + 11-s − 0.569·12-s + 13-s − 2.94·15-s − 4.90·16-s + 2.92·17-s − 3.82·18-s − 3.41·19-s + 2.49·20-s + 1.64·22-s − 4.64·23-s + 1.75·24-s + 7.87·25-s + 1.64·26-s + 4.36·27-s − 2.28·29-s − 4.82·30-s − 4.28·31-s − 3.77·32-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.473·3-s + 0.347·4-s + 1.60·5-s − 0.549·6-s − 0.757·8-s − 0.776·9-s + 1.86·10-s + 0.301·11-s − 0.164·12-s + 0.277·13-s − 0.759·15-s − 1.22·16-s + 0.710·17-s − 0.901·18-s − 0.783·19-s + 0.558·20-s + 0.350·22-s − 0.969·23-s + 0.358·24-s + 1.57·25-s + 0.321·26-s + 0.840·27-s − 0.424·29-s − 0.881·30-s − 0.768·31-s − 0.667·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.634516346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634516346\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 3 | \( 1 + 0.819T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 + 3.41T + 19T^{2} \) |
| 23 | \( 1 + 4.64T + 23T^{2} \) |
| 29 | \( 1 + 2.28T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 8.72T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 - 8.52T + 61T^{2} \) |
| 67 | \( 1 + 6.03T + 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 + 0.442T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
| 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
−7.892014415096303088309649126681, −6.79542262992294238563730471148, −6.07679648901536750808623594422, −5.77503187911670891013734918677, −5.41959015478651915356247718119, −4.41335367276444589887766120032, −3.76650682675867415236805229543, −2.65872221812241333850852940300, −2.18970970173457663745218005823, −0.826785888294024181758719108524,
0.826785888294024181758719108524, 2.18970970173457663745218005823, 2.65872221812241333850852940300, 3.76650682675867415236805229543, 4.41335367276444589887766120032, 5.41959015478651915356247718119, 5.77503187911670891013734918677, 6.07679648901536750808623594422, 6.79542262992294238563730471148, 7.892014415096303088309649126681
