| L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s + 11-s + 2.82·13-s + 16-s + 5.65·17-s + 2.82·19-s − 1.41·20-s + 22-s + 2·23-s − 2.99·25-s + 2.82·26-s + 6·29-s − 1.41·31-s + 32-s + 5.65·34-s − 2·37-s + 2.82·38-s − 1.41·40-s − 5.65·41-s + 4·43-s + 44-s + 2·46-s − 4.24·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s + 0.301·11-s + 0.784·13-s + 0.250·16-s + 1.37·17-s + 0.648·19-s − 0.316·20-s + 0.213·22-s + 0.417·23-s − 0.599·25-s + 0.554·26-s + 1.11·29-s − 0.254·31-s + 0.176·32-s + 0.970·34-s − 0.328·37-s + 0.458·38-s − 0.223·40-s − 0.883·41-s + 0.609·43-s + 0.150·44-s + 0.294·46-s − 0.618·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.481547169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.481547169\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + 4.24T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
| 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
−7.52439934963764878974639506515, −7.05320983189886319611053331847, −6.16945730938830495618789705776, −5.63771500837814560750558809974, −4.89129234534473132451797989846, −4.12104537381210029049728139367, −3.44978499480081786200908564789, −2.96492979434940412659366783048, −1.71120259922571551647729693970, −0.838545002784852569998979181643,
0.838545002784852569998979181643, 1.71120259922571551647729693970, 2.96492979434940412659366783048, 3.44978499480081786200908564789, 4.12104537381210029049728139367, 4.89129234534473132451797989846, 5.63771500837814560750558809974, 6.16945730938830495618789705776, 7.05320983189886319611053331847, 7.52439934963764878974639506515
