L(s) = 1 | + 2·2-s + 7.28·3-s + 4·4-s − 5·5-s + 14.5·6-s + 8·8-s + 26.0·9-s − 10·10-s + 41.1·11-s + 29.1·12-s + 24.5·13-s − 36.4·15-s + 16·16-s + 93.3·17-s + 52.0·18-s − 54.6·19-s − 20·20-s + 82.2·22-s − 136.·23-s + 58.2·24-s + 25·25-s + 49.1·26-s − 7.11·27-s + 282.·29-s − 72.8·30-s + 54.6·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.447·5-s + 0.990·6-s + 0.353·8-s + 0.963·9-s − 0.316·10-s + 1.12·11-s + 0.700·12-s + 0.524·13-s − 0.626·15-s + 0.250·16-s + 1.33·17-s + 0.681·18-s − 0.659·19-s − 0.223·20-s + 0.796·22-s − 1.23·23-s + 0.495·24-s + 0.200·25-s + 0.370·26-s − 0.0507·27-s + 1.81·29-s − 0.443·30-s + 0.316·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.059980351\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.059980351\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 7.28T + 27T^{2} \) |
| 11 | \( 1 - 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 193.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 419.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 57.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 687.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 716.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.83e3T + 9.12e5T^{2} \) |
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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
−10.46581269500863730975086251245, −9.605488697937776392034046870569, −8.502142414032871905314937158410, −8.030169891647002584456455529979, −6.90572434328813756458729067610, −5.92815251959581131435959394011, −4.37138996110533049910261874085, −3.67489314831101197136114357686, −2.75816177481517482271540844081, −1.39154921484711168005347351228,
1.39154921484711168005347351228, 2.75816177481517482271540844081, 3.67489314831101197136114357686, 4.37138996110533049910261874085, 5.92815251959581131435959394011, 6.90572434328813756458729067610, 8.030169891647002584456455529979, 8.502142414032871905314937158410, 9.605488697937776392034046870569, 10.46581269500863730975086251245