L(s) = 1 | + (2.5 + 0.866i)7-s + 5·13-s + (0.5 + 0.866i)19-s + (2.5 − 4.33i)25-s + (−5.5 + 9.52i)31-s + (−5.5 − 9.52i)37-s − 13·43-s + (5.5 + 4.33i)49-s + (−7 − 12.1i)61-s + (−2.5 + 4.33i)67-s + (−8.5 + 14.7i)73-s + (−8.5 − 14.7i)79-s + (12.5 + 4.33i)91-s + 14·97-s + (6.5 + 11.2i)103-s + ⋯ |
L(s) = 1 | + (0.944 + 0.327i)7-s + 1.38·13-s + (0.114 + 0.198i)19-s + (0.5 − 0.866i)25-s + (−0.987 + 1.71i)31-s + (−0.904 − 1.56i)37-s − 1.98·43-s + (0.785 + 0.618i)49-s + (−0.896 − 1.55i)61-s + (−0.305 + 0.529i)67-s + (−0.994 + 1.72i)73-s + (−0.956 − 1.65i)79-s + (1.31 + 0.453i)91-s + 1.42·97-s + (0.640 + 1.10i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39091 + 0.0882674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39091 + 0.0882674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (5.5 - 9.52i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (8.5 - 14.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−11.99929450501712945365765238435, −11.08157249016867279069507421164, −10.37132101945693126350887489407, −8.891602898263775853725812511253, −8.372007332579292663194765642537, −7.12169118176407111083321705790, −5.90699610136419423916435487577, −4.84025151217890051943393771530, −3.46639878577192253208900107072, −1.67259348408527152003717601706,
1.54297146000919957214995670622, 3.46622239353313215930627619790, 4.71978140179157600387267003743, 5.89035506414127979385176340765, 7.13029545854197214861403933751, 8.172325327717420668150147378323, 8.989685743507978045566270894294, 10.26770479503104807237374766039, 11.16649553244544186493239264889, 11.76781945541260148277579085538