L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 28.1i·5-s − 18.5·7-s + 22.6i·8-s − 79.7·10-s + 168. i·11-s + 319.·13-s + 52.3i·14-s + 64.0·16-s + 39.1i·17-s + 457.·19-s + 225. i·20-s + 476.·22-s − 435. i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.12i·5-s − 0.377·7-s + 0.353i·8-s − 0.797·10-s + 1.39i·11-s + 1.88·13-s + 0.267i·14-s + 0.250·16-s + 0.135i·17-s + 1.26·19-s + 0.563i·20-s + 0.984·22-s − 0.822i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.982679539\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.982679539\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 18.5T \) |
good | 5 | \( 1 + 28.1iT - 625T^{2} \) |
| 11 | \( 1 - 168. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 319.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 39.1iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 457.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 435. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 135. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 267.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 812.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 737. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.09e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.19e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.62e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.86e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.00e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.30e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.31e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.70e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 9.03e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 3.71e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.37e4T + 8.85e7T^{2} \) |
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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
−10.47309855734118655210746670320, −9.581532019279355264587961675426, −8.859234922150909805820291736171, −7.998465798180781423953243244353, −6.63513797375301590171830836386, −5.37745397214819855024757445209, −4.44300102444103114304360814004, −3.40177278654584788309302066770, −1.78069874509891056203252195031, −0.78281982667495288420728384043,
0.978598553003939144593399100929, 3.10856963233416816976249377835, 3.71591298352670983082474059835, 5.55951633906580789875979502021, 6.19691577013487179154722082794, 7.06620695377778370016465405782, 8.112587068965606261063546253471, 8.968160541574592945990762191030, 10.01968629501746567321510718484, 11.06485699551796229417757424753