| L(s) = 1 | + 2-s + 4-s + 4.26·5-s − 1.12·7-s + 8-s + 4.26·10-s − 5.86·11-s + 0.598·13-s − 1.12·14-s + 16-s + 3.29·17-s + 4.82·19-s + 4.26·20-s − 5.86·22-s + 3.67·23-s + 13.1·25-s + 0.598·26-s − 1.12·28-s + 2.48·29-s + 4.00·31-s + 32-s + 3.29·34-s − 4.78·35-s − 5.08·37-s + 4.82·38-s + 4.26·40-s + 4.82·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.90·5-s − 0.423·7-s + 0.353·8-s + 1.34·10-s − 1.76·11-s + 0.165·13-s − 0.299·14-s + 0.250·16-s + 0.799·17-s + 1.10·19-s + 0.953·20-s − 1.25·22-s + 0.765·23-s + 2.63·25-s + 0.117·26-s − 0.211·28-s + 0.461·29-s + 0.719·31-s + 0.176·32-s + 0.565·34-s − 0.808·35-s − 0.835·37-s + 0.783·38-s + 0.674·40-s + 0.753·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.459011468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.459011468\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 - 0.598T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 4.00T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 0.626T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 - 9.61T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 + 9.66T + 61T^{2} \) |
| 67 | \( 1 + 8.86T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 0.687T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 + 4.41T + 89T^{2} \) |
| 97 | \( 1 + 5.36T + 97T^{2} \) |
| 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
−9.788616140070876534510175284896, −8.846715007154415916424768755916, −7.75018094730587472403293864252, −6.87059595384933289545043317398, −5.95522314867867893338507741270, −5.42573546609644535437216636588, −4.83066554950821293416626631859, −3.08764407405529952461271954510, −2.64985701047282270936698107438, −1.37028910150554257092518458269,
1.37028910150554257092518458269, 2.64985701047282270936698107438, 3.08764407405529952461271954510, 4.83066554950821293416626631859, 5.42573546609644535437216636588, 5.95522314867867893338507741270, 6.87059595384933289545043317398, 7.75018094730587472403293864252, 8.846715007154415916424768755916, 9.788616140070876534510175284896
