L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s − 1.80·5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s + 0.801·10-s − 12-s − 2.24·15-s + 0.445·16-s − 0.246·18-s − 0.445·19-s + 1.44·20-s + 24-s + 2.24·25-s − 0.554·27-s + 1.24·29-s + 0.999·30-s − 32-s − 0.445·36-s − 0.445·37-s + 0.198·38-s − 1.44·40-s − 1.80·43-s − 0.999·45-s + 0.554·48-s + ⋯ |
L(s) = 1 | − 0.445·2-s + 1.24·3-s − 0.801·4-s − 1.80·5-s − 0.554·6-s + 0.801·8-s + 0.554·9-s + 0.801·10-s − 12-s − 2.24·15-s + 0.445·16-s − 0.246·18-s − 0.445·19-s + 1.44·20-s + 24-s + 2.24·25-s − 0.554·27-s + 1.24·29-s + 0.999·30-s − 32-s − 0.445·36-s − 0.445·37-s + 0.198·38-s − 1.44·40-s − 1.80·43-s − 0.999·45-s + 0.554·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4178985571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4178985571\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 - T \) |
good | 2 | \( 1 + 0.445T + T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.80T + T^{2} \) |
| 79 | \( 1 + 1.80T + T^{2} \) |
| 83 | \( 1 + 0.445T + T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.92707055095013819463169451241, −14.03910777988174726171598133235, −12.88778415189161766638407934767, −11.67985104016781996899636064300, −10.24785144415084892602116520791, −8.775621347957351956553642935712, −8.308375254782647564178629435651, −7.32935458632589252277476848657, −4.51844117616935213608968885293, −3.39979849702102685652285544820,
3.39979849702102685652285544820, 4.51844117616935213608968885293, 7.32935458632589252277476848657, 8.308375254782647564178629435651, 8.775621347957351956553642935712, 10.24785144415084892602116520791, 11.67985104016781996899636064300, 12.88778415189161766638407934767, 14.03910777988174726171598133235, 14.92707055095013819463169451241