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 | \( 1 + 14T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.733484805010312995757072075307, −8.647372999198639884868872893249, −8.080852764483335655557793315926, −6.99228793999434855221368045232, −6.61471505043786677071393997187, −5.19829198354000330994047648204, −4.88594222999733455436622366586, −3.56340001883532626000430154013, −2.68442837366993884113119511113, −1.43413597535384233558640062636,
0.33433785379799074607024525153, 2.03418938979461441867474811686, 2.90917606892586871300613969247, 4.14643365204930599716803810902, 4.89580628301305627995229613751, 5.79784314409085780556131012788, 6.75045881725246026399194981917, 7.44085467768578459224412039613, 8.255530357163612598956641288302, 9.065661880740095282906361013185