L(s) = 1 | − 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 + (7 + 12.1i)61-s + (−2.5 + 4.33i)67-s + (8.5 − 14.7i)73-s + (−8.5 − 14.7i)79-s − 14·97-s + (−6.5 − 11.2i)103-s + (9.5 − 16.4i)109-s + ⋯ |
L(s) = 1 | − 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 + (0.896 + 1.55i)61-s + (−0.305 + 0.529i)67-s + (0.994 − 1.72i)73-s + (−0.956 − 1.65i)79-s − 1.42·97-s + (−0.640 − 1.10i)103-s + (0.909 − 1.57i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9123965823\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9123965823\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
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
−9.065661880740095282906361013185, −8.255530357163612598956641288302, −7.44085467768578459224412039613, −6.75045881725246026399194981917, −5.79784314409085780556131012788, −4.89580628301305627995229613751, −4.14643365204930599716803810902, −2.90917606892586871300613969247, −2.03418938979461441867474811686, −0.33433785379799074607024525153,
1.43413597535384233558640062636, 2.68442837366993884113119511113, 3.56340001883532626000430154013, 4.88594222999733455436622366586, 5.19829198354000330994047648204, 6.61471505043786677071393997187, 6.99228793999434855221368045232, 8.080852764483335655557793315926, 8.647372999198639884868872893249, 9.733484805010312995757072075307