L(s) = 1 | − 2-s + (1.72 + 0.115i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1.72 − 0.115i)6-s + (1.23 − 2.33i)7-s − 8-s + (2.97 + 0.400i)9-s + (0.5 + 0.866i)10-s + (1.85 − 3.21i)11-s + (1.72 + 0.115i)12-s + (−1.26 + 2.19i)13-s + (−1.23 + 2.33i)14-s + (−0.763 − 1.55i)15-s + 16-s + (−1.48 − 2.56i)17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.997 + 0.0669i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.705 − 0.0473i)6-s + (0.468 − 0.883i)7-s − 0.353·8-s + (0.991 + 0.133i)9-s + (0.158 + 0.273i)10-s + (0.559 − 0.969i)11-s + (0.498 + 0.0334i)12-s + (−0.351 + 0.609i)13-s + (−0.331 + 0.624i)14-s + (−0.197 − 0.401i)15-s + 0.250·16-s + (−0.359 − 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37775 - 0.671326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37775 - 0.671326i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.72 - 0.115i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.23 + 2.33i)T \) |
good | 11 | \( 1 + (-1.85 + 3.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.26 - 2.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.48 + 2.56i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.26 - 2.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.21 + 2.10i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.04 + 8.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 + (-1.36 + 2.37i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.50 - 9.53i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.65 - 2.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 0.695T + 47T^{2} \) |
| 53 | \( 1 + (-4.25 - 7.37i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 2.43T + 71T^{2} \) |
| 73 | \( 1 + (-0.729 - 1.26i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 + (5.39 + 9.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.50 - 13.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.0494 - 0.0856i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.15873907014077142026975223008, −9.576667381982433317111651261252, −8.578746602768917489279994557326, −8.095272149313059941250851168663, −7.23010102292413917914135299022, −6.27171831306316953364707690763, −4.58240243746109611402564578949, −3.81105826471799135147906750983, −2.41367702432752893146714146140, −1.02566870571423921519659894992,
1.75074932614824421364297080782, 2.65802543077249008264480254044, 3.90809011931793861684018202564, 5.21363800798102389169561134418, 6.67198591147576454197958399140, 7.32057537822063420140557243228, 8.332019744585667318476820031561, 8.810661982012815438844999107036, 9.767556862267773803399412230589, 10.42948095717764522513083791080