| L(s) = 1 | + 1.56·2-s + 0.460·4-s + 4.07·5-s − 2.41·8-s + 6.38·10-s + 4.60·11-s − 13-s − 4.70·16-s + 3.53·17-s − 5.76·19-s + 1.87·20-s + 7.21·22-s + 4.38·23-s + 11.5·25-s − 1.56·26-s − 4.06·29-s + 3.28·31-s − 2.55·32-s + 5.54·34-s + 8.96·37-s − 9.04·38-s − 9.83·40-s + 1.14·41-s − 7.33·43-s + 2.11·44-s + 6.87·46-s + 7.46·47-s + ⋯ |
| L(s) = 1 | + 1.10·2-s + 0.230·4-s + 1.82·5-s − 0.853·8-s + 2.01·10-s + 1.38·11-s − 0.277·13-s − 1.17·16-s + 0.858·17-s − 1.32·19-s + 0.418·20-s + 1.53·22-s + 0.914·23-s + 2.31·25-s − 0.307·26-s − 0.755·29-s + 0.589·31-s − 0.451·32-s + 0.951·34-s + 1.47·37-s − 1.46·38-s − 1.55·40-s + 0.178·41-s − 1.11·43-s + 0.319·44-s + 1.01·46-s + 1.08·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.069216785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.069216785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 + 4.06T + 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 - 7.46T + 47T^{2} \) |
| 53 | \( 1 + 4.15T + 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 - 0.567T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 1.93T + 73T^{2} \) |
| 79 | \( 1 - 3.44T + 79T^{2} \) |
| 83 | \( 1 - 8.69T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
| 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
−8.218514735069598813022612075031, −6.93751538047433335607662880112, −6.40958006257169039523346134481, −5.93072075666671640094720237788, −5.26978602899564885924240699449, −4.57023295081950182546924574476, −3.78165715031419844821281378803, −2.84429751057442105604827604911, −2.09235221339557858511644717270, −1.08357482363150644999794503020,
1.08357482363150644999794503020, 2.09235221339557858511644717270, 2.84429751057442105604827604911, 3.78165715031419844821281378803, 4.57023295081950182546924574476, 5.26978602899564885924240699449, 5.93072075666671640094720237788, 6.40958006257169039523346134481, 6.93751538047433335607662880112, 8.218514735069598813022612075031
