| L(s) = 1 | + 2-s + 2.24·3-s + 4-s + 1.55·5-s + 2.24·6-s + 3.49·7-s + 8-s + 2.04·9-s + 1.55·10-s − 3.74·11-s + 2.24·12-s − 3.85·13-s + 3.49·14-s + 3.49·15-s + 16-s + 4.93·17-s + 2.04·18-s + 6.15·19-s + 1.55·20-s + 7.85·21-s − 3.74·22-s − 5.76·23-s + 2.24·24-s − 2.58·25-s − 3.85·26-s − 2.13·27-s + 3.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.695·5-s + 0.917·6-s + 1.32·7-s + 0.353·8-s + 0.682·9-s + 0.491·10-s − 1.12·11-s + 0.648·12-s − 1.06·13-s + 0.933·14-s + 0.902·15-s + 0.250·16-s + 1.19·17-s + 0.482·18-s + 1.41·19-s + 0.347·20-s + 1.71·21-s − 0.797·22-s − 1.20·23-s + 0.458·24-s − 0.516·25-s − 0.755·26-s − 0.411·27-s + 0.660·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.923152857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.923152857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 5 | \( 1 - 1.55T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 - 6.15T + 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 + 0.0271T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 0.554T + 47T^{2} \) |
| 53 | \( 1 - 3.25T + 53T^{2} \) |
| 59 | \( 1 + 3.06T + 59T^{2} \) |
| 61 | \( 1 + 0.131T + 61T^{2} \) |
| 67 | \( 1 + 11T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 - 6.62T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 + 4.31T + 97T^{2} \) |
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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
−9.436253820321982142678948813552, −8.343671427229291284966682989418, −7.62569224962550377882452005503, −7.43959448582500253927460332165, −5.65161625202525762540718433678, −5.37680658228399008509620060644, −4.32359227633140756332062589567, −3.21629652050587479329959357881, −2.41494598642847298886400091075, −1.66981753705387663269958363527,
1.66981753705387663269958363527, 2.41494598642847298886400091075, 3.21629652050587479329959357881, 4.32359227633140756332062589567, 5.37680658228399008509620060644, 5.65161625202525762540718433678, 7.43959448582500253927460332165, 7.62569224962550377882452005503, 8.343671427229291284966682989418, 9.436253820321982142678948813552
